Exactly What It Says on the Tin--a church located in America.
...Ohhhh, you mean an official church of the United States of America.
There isn't one.
The First Amendment to the Constituion of the United States begins Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof...
In the United States, the First Amendment puts an extremely strong prohibition on the government regulating religion or endorsing one, up to and specifically forbidding a government church. This is known as the principle of "separation of church and state." Now, this doesn't give churches unlimited freedom to do anything they want -- church buildings still have to comply with building codes (although they're generally exempt from height limitations), and anyone claiming something ridiculous and illegal (like, say, Human Sacrifice) as part of their religion will be laughed out of court (and in the case of Human Sacrifice, promptly convicted of murder). And while the involvement of religious organizations in politics isn't forbidden by the Constitution, incorporated churches can lose their tax-exempt status if they do so (as they would then be considered a political lobbying group), causing them to keep a low profile in politics. Nearly all religious-based lobbying is done by non-profit groups for exactly this reason.
The government can't decide that your religion is unworthy, isn't right, or is a cult simply because people think that it's heretical or blasphemous. To do that, they will go after something else: too many guns and paedophiles at Waco, too much polygamy and forced marriages of young girls to older men at that Fundamentalist Mormon compound in Utah. But if a bunch of adults decide to hold Satanic services involving devil worship, short of finding something actually illegal going on, there ain't a damn thing the government can do to stop it. So if you want to start a cult that says the world is cube-shaped and your deity is a talking lizard, you're A-OK.
A government agency, state or federal, can deny a permit to operate virtually any business, be it a pawn shop, a liquor store, or even a non-profit operation like a thrift store or a charity, but by law, it can't even require a church to have a permit. You might not be able to invite 10 people into your home for a Tupperware party, due to hard rules on commercial activity, but you can always invite 10 people over for a prayer meeting. Churches are also exempt from taxation, as this is considered a violation of the separation of church and state (although see above for one of the big exceptions). Thus, churches in the United States have a great deal of power in the way they operate themselves without fear of the government shutting them down, unlike, say, Falun Gong in China or a Christian church operating in Saudi Arabia.
A commonly-held view among Europeans and Canadians is that Americans are far more religious than the rest of the Western world, and to a great extent, this view is Truth in Television. There are some areas of the country (most notably New England, the Pacific Northwest and most of the Rocky Mountain region) that have rates of religiosity more in line with Canadian or European norms, but when one is in, say, the Deep South, the rural Midwest or Utah, it's not as much of a joke as a non-native would think to say that the church is the local government, and vice versa. Such areas tend to have firmly established churches that are heavily integrated into the local community and are often a major part of community life. It's very common, especially in small rural towns, for churches to be the center of the community, and for everyone in that town, from the mayor to convenience store owners, to attend church on Sunday. This has a profound effect on the prevailing social views within the US; it explains why there exists more resistance to things like abortion, homosexuality and the teaching of evolution than in some other countries, as well as the relative strictness of the Media Watchdogs at the FCC, especially on matters of sex and profanity.
There was a skit on Rowan and Martin's Laugh-In that showcased the leeway that churches have. A county tax assessor visits a small grocery store to examine the place and estimate its value for property tax purposes. The owner complains about how much his property taxes keep going up, and the assessor notes the big problem is all of the churches in the area that don't pay property taxes. The owner gets an idea, and as the assessor is leaning over the front counter next to the cash register to write up some notes, the owner says to him, "Stop leaning on my altar!"
In the United States, Christian churches tend to fall into the following groups:
- The Roman Catholic Church is the largest single denomination in not only the United States, but the world (although Sunni Islam is catching up with the second one). Historically, Catholicism was the religion of The City in general, and of immigrant ethnic groups (Irish, Italians, Poles and, more recently, Latinos) in particular. The association with immigrants and the supposed decadence of big cities, combined with America's longstanding Protestant tradition, led to widespread anti-Catholic prejudice in the 19th and early 20th centuries, with many claiming that the Catholics were agents of The Pope who were trying to subvert American society. However, outside of a few diehard fundamentalists (such as Jack Chick), this attitude has mostly disappeared, the pivotal moment being the election of John F. Kennedy as America's first Catholic President in 1960. Historically, Catholics have generally been concentrated in the Northeast, Midwest, and in Louisiana. More recently, a mix of Latino immigration and internal migration has made the Church more popular in California, the Southwest, and Florida.
American Catholics are often perceived as being more liberal than the American mainstream, dissenting from the Church in Rome on many social/cultural issues (such as gay rights, allowing priests to marry, birth control, and the ordination of women and gays) while supporting activism for social justice projects. However, the culturally conservative direction taken by the Church in the last few decades under Popes John Paul II and Benedict XVI means that many recent converts tend to have more conservative views on social issues than the stereotype suggests. The growing Latino contingent within American Catholicism also tends to hold more traditional values.
- Mainline Protestant churches include the Methodists, the Lutherans, the Presbyterians, the Episcopalians (the American branch of the Anglicans), the United Church of Christ and other well-established, or "heritage", churches. These churches are the more liberal of the two Protestant groups, and will often take moderate or liberal positions on social issues. They are generally concentrated in the Northeast and the Midwest. Lately, these churches have been seeing steep declines in membership as people gravitate to either the more conservative evangelical churches or toward more secular outlooks. At the same time, some of the churches have been battling conservative defections due to their liberal social positions, especially with regards to homosexuality. The Episcopal Church's election of an openly gay bishop, for example, caused some parishes to break away and align themselves with more conservative Anglican denominations in Africa.
- Evangelical churches, as defined by The Other Wiki, are Protestant churches that are distinguished by four key traits -- a focus on personal conversion (becoming "born again"), spreading the message of The Bible (evangelizing), placing high stock in Biblical authority, and a focus on Jesus' death and resurrection. Examples of such churches include most subgroups of Baptists, the Missouri Synod Lutherans, and the Presbyterian Church of America . They usually adhere to conservative social values, and are very often fundamentalist. 
They are the largest of the major Christian groups in America, and are most heavily concentrated in an area known as the "Bible Belt", consisting of the South, Texas and parts of the Midwest. They have grown quite strongly in recent years, due to their focus on missionary activity. If a character is described as a "born-again Christian," then he or she is most likely an evangelical -- the two terms are largely seen as interchangeable in American usage. Lately, they have become the stereotypical American churches.
They are also responsible for the growth of what are often called megachurches. While a more traditional church will have from a few dozen to a few hundred parishioners return every week, with "extracurricular" services largely limited to Sunday schools, bake sales and grade schools for some of the larger ones, a megachurch has a few thousand or even tens of thousands, and its services will often be more comparable to a rock concert than an old-time congregation. Megachurches are likely to have their own K-12 schools, fitness centers, day cares, shops selling Christian merchandise (some of it likely pertaining to, or created by, the head pastor/minister), and ministries targeting various subcultures, making them one-stop shops for born-again suburbanites. The trend began in the middle of the twentieth century and is associated with the rise of the Religious Right and the growth of the evangelical and Pentecostal movements, as they tend to focus on conversion and personal morality/salvation. These churches have been the target of criticism by both Christians and non-Christians alike, for drawing parishioners away from traditional churches, for their "big box" feel and perceived focus on consumerism, their use of secular business models to bring in worshipers and dollars, and their tax-exempt status.
- Related to the Evangelical movement is Pentecostalism, which writers often lump in with the evangelicals due to a lack of familiarity with either one. Pentecostalism is based around a direct experience with God, and often includes faith healing, speaking in tongues, and getting "imbued" with the Holy Spirit. Often associated with loud, charismatic preachers (indeed, a subset of Pentecostalism is called the "charismatic movement," although the word has a more specific meaning in this context), who many skeptics will claim are responsible for the activity that goes on during Pentecostal sermons due to their getting the crowd riled up.
- Despite the fact that Hollywood screenwriters often lump Pentecostals and evangelicals together (most likely due to their shared social conservatism), the two groups differ on a great number of theological issues, which has led to some friction between them. Pentecostals and Charismatics believe in faith healing, speaking in tongues, and a continuing tradition/gift of prophecy (think mystics). Evangelicals believe in personal revelation and experience. Both believe in biblical inerrancy/literalism and baptism/rebirth in Christ. Pat Robertson (Charismatic) makes doomsday predictions and believes he speaks with the voice of God. Evangelicals confine themselves to agreeing with him when, ex post facto, he says a certain city was punished for not punishing homosexuality. To confuse things further, there are charismatic Catholics, too.
- The Orthodox churches include the Eastern Orthodox (Russian Orthodox, Greek Orthodox, etc.) and Oriental Orthodox (Coptic Orthodox, Armenian Apostolic, Ethiopian Orthodox, etc).  They are both descended from the Orthodox Church of the Byzantine Empire, which split with the Roman Catholic Church in the 11th century following disputes over the respective authority of the Pope versus the Eastern Roman Emperor and Patriarch of Constantinople, doctrinal disputes over liturgy and the use of icons, and just plain West vs. East bigotry. The two, Pope and Patriarch, mutually excommunicated each others' followers around 1055.
Orthodox Christians make up less than 1% of the American population, and are associated with particular ethnic groups even more than the Catholics. Basically, if someone is from Eastern Europe or the Balkans, they're more likely than not Eastern Orthodox, and vice versa. Copts aren't as common in America as they are in Canada. They're mostly from Egypt, Turkey, and Syria. There are purportedly 700,000 to 1 million Copts in America, but in very clustered communities.  The most common denominations of Oriental Orthodox in America are the Armenian Apostolic (mostly in California) and Ethiopian Orthodox (mostly in the Midwest) churches.
- The Church of Jesus Christ of Latter-day Saints (or the LDS Church, or simply the Mormons) is a Nontrinitarian orientation (which basically means they don't believe in the Holy Trinity), and is based mostly in the state of Utah, although there are also significant populations in California, Idaho, Nevada and Arizona. They are stereotyped as having deeply conservative social views -- a view that is usually Truth in Television. Owing to their history of persecution and hard life on the frontier, they also believe in self-sufficiency -- the Church recommends that all Mormons keep a few months of supplies on hand, and the Church itself has a massive stockpile of food and supplies in the event of a disaster. They also put more emphasis on helping people find jobs and become self-sufficient than on handing out charity or operating soup kitchens.
Their religious beliefs often conflict with those mainstream Christianity, particularly with regard to their belief in The Book of Mormon, which they believe to be a holy text on par with the Gospels. This, combined with their past practice of polygamy (which is not helped by the existence of breakaway sects that still practice it, in violation of both the law and current LDS Church doctrine), means that they are still an acceptable target in many parts of the country, from both conservatives who view them as a Cult and secularists who associate them with the rest of the Christian Right. This became evident in the 2008 and 2012 Republican primaries, when Mitt Romney's Mormon faith caused issue with some Christian conservatives, and in the California Proposition 8 debate in 2008, going by some of the reactions by the anti-Prop 8 side to their influence over the gay-rights debate in California.
- Jehovah's Witnesses, like the LDS Church, are Nontrinitarian, evangelical, and conservative, and are known to come off as strange to the majority of Americans. They are infamous for their door-to-door preaching and proselytizing (so much that it even got them their own trope), and they keep track of how much time they spend in those activities, trying to be the most passionate and zealous missionaries they can possibly be. They don't observe Christmas, Easter, or birthdays, which are pagan in origin, or national holidays like Thanksgiving or Independence Day. However, they do celebrate the Lord's Evening Meal, held on Passover, which is similar to Eucharist, but they don't believe in transubstantiation or consubstantiation. They do not participate in the military or warfare in general and refuse to salute national flags, which has gotten them in lots of trouble (especially in public schools, what with the Pledge of Allegiance). They're also famous for refusing to use certain blood products, even if they're dying. This means no blood transfusions or emergency surgery that requires transfusions of blood or blood products from another person. Finally, they feel that The End of the World as We Know It is imminent, and have, in the past, tried to pin down the exact date of the Apocalypse. They stopped doing this when they realize that it was earning them more mockery than converts, but eschatology is still a major part of their belief system.
- Much less common than the above, but still prevalent in America, are Anabaptists. They are the descendants of the Radical Reformation, alongside the Protestant Reformation, who believe that being baptized and joining a church should be a choice, offered only to adults who had the knowledge to make such a decision. This sounds sensible now, but was pretty revolutionary in the 16th century.
Surprisingly, this is not some liberal hippie denomination, but actually the "plain people": Amish, Hutterites, and Mennonites. Each group is distinct from the others, but they all share core beliefs. They're famous for being actual pacifists (believing that Turn the Other Cheek isn't just a suggestion), and also refusing to swear oaths, participate in politics, or drink any alcohol. The more conservative groups dress in plain clothes, keep technology use to a minimum, live in their own separate communities, and refuse to pay Social Security or even for insurance. More moderate branches, particularly the Mennonites, blend in more with society, though they still stick to their theology. Unusually for a Christian group, Anabaptists have their own language: Low German, also known as Plautdietsch.
- The Religious Society of Friends, or Quakers (the term was originally applied as an insult, but the Friends embraced it) is not so much a sect as a (very) loose network of people who agree on certain principles, and agree to disagree on others. There is enough variety and universalism among the Friends that a significant minority of them do not identify as Christian. The Friends have no clergy, and no creed, and are as a whole much more interested in activism and social justice than proselytizing.
Other religions are also well-represented in the country, although all of them are clear minorities of the United States population.
- Jews are primarily concentrated on the East Coast (particularly the New York and Washington areas, where they make up a double-digit percentage of the population in some counties), South Florida (where many of them go to retire), and California, with small enclaves elsewhere in the country. Their disproportionate presence in New York and Los Angeles explains why Jews are so well-represented in the entertainment and financial industries. They tend to pull for the Democrats in large numbers -- the only Democrat since FDR who earned less than 70% of the Jewish vote was Adlai Stevenson in 1952 and 1956 -- and usually have social views more liberal than the American mainstream.  In America, as in most other places, Jews are Once-Acceptable Targets -- anti-Semitism was prevalent in America as late as The Great Depression (during which time populist radio host Father Coughlin blamed the Jews for the stock market crash), but slowly began to fade after the atrocities of World War II were brought to the surface.
- There is significant debate of the number of Muslims in the United States, with most estimates ranging from as low as one million to as high as seven million. Two-thirds of the Muslim community is foreign-born, while most of the rest consists of African-American converts (where we get the stereotypical "black Muslims"). Almost one-fifth of American convicts are Muslims, most of whom converted to Islam while in prison (again, the "black Muslim" stereotype). They tend to be concentrated on the East Coast, in Detroit, in Houston, and in California.
Thanks to the 9/11 attacks, Muslims have become acceptable targets in many parts of the United States, which has caused many of them to support the Democratic Party.  Even so, they tend to be rather well-integrated compared to their European counterparts, having a higher average income and educational attainment than the national average. The first Muslim member of Congress is Keith Ellison, an African-American Democrat from Minnesota elected in 2006; he was followed by another African American, Andre Carson (Democrat of Indiana), after a special election in 2008. And no, despite what you may have heard, President Barack Obama is not a Muslim (Not That There Would Be Anything Wrong With That...).
- The Nation of Islam, whose more prominent members have included Malcolm X and Louis Farrakhan, and Muhammad Ali, is an American offshoot of the religion of Islam. As with mainstream Islam, the NOI preaches adherence to the five pillars of Islam, personal modesty, eschewing pork, and many other similarities. They differ from mainstream Islam in that they also preach black supremacy and that their founder, Wallace Fard Muhammad, was the Christian Messiah and the Muslim Mahdi (much the same thing). With its own religious text, doctrinal differences with traditionalists, and American origin, the Nation of Islam can be seen as analogous to Mormonism, and its reception has often been similar (frosty at best, hostile at worst).
- Atheists and other non-religious people (not a church, but here for completeness) make up about 15% of the American population, with their numbers being highest in the Western states and the Northeast. The vast majority of non-religious people tend to be either liberals (they voted about 71% for Barack Obama) or libertarians, although there are a few prominent conservative atheists. They have long been an acceptable target in American culture, often being stereotyped as bitter, elitist, amoral, un-patriotic, and possibly Communist. However, lately there have been some more positive portrayals of non-religious people in the media, such as Temperance Brennan and many of Joss Whedon's (himself an atheist) characters.
- Agnostics are people who are open to the idea of an afterlife and/or a higher power, but who don't ascribe to any one religion. Tend to get lumped in with atheists by some religious communities, despite the difference. Most common in the Northeast and the Pacific Northwest.
- Buddhists make up 1-2% of the American population. About 75-80% of American Buddhists are Asian, while most of the rest are white converts (although, as noted below, not entirely white). This latter group is typically stereotyped as consisting largely of New Age Retro Hippies and Granola Girls, while the former group is usually stereotyped as... well, Asian. There have been precisely three Buddhists in Congress, all Democrats: Mazie Hirono, a non-practicing Issei Japanese-American from Hawaii, and Hank Johnson, a Black convert from Georgia (yeah, it's kind of weird) were elected in 2006; Colleen Hanabusa, a practicing Yonsei Japanese-American from Hawaii was elected in 2010.
- There are about 1-1.5 million Hindus in the United States. Most of them are South Asian immigrants who, like their East Asian Buddhist counterparts, have their own stereotypes (such as the Asian Store Owner and the Bollywood Nerd). Hindu gurus had a large influence in the American New Age movement in The Sixties and The Seventies, when it attracted a number of high-profile Western converts. Most American perceptions of the religion stem from this, and from what is gleaned of the Indian community.
- There are between 1 and 1.2 million neo-pagans in the United States. The largest neo-pagan denomination is Wicca, which is estimated as having between 150,000 and 300,000 followers. They are typically seen as acceptable targets, due to the fact that they practice a faith claiming lineage from the pagan, pre-Christian religions of Europe. As such, they have often been falsely accused of Satanism and criminal activity (including child abuse and sacrifice) by conservative Christians (then again, conservative Christians consider ALL other religions to be (unwitting at the very least) Satanism, except for Judaism which they consider pre-Christianity), although this has declined in recent years.
- While Wicca is the predominant neopagan faith, there are plenty of others, though they have even smaller representations:
- Alchemy is a reconstruction of alchemical ideals reimaged as spiritual descriptions rather than practical chemistry.
- Asatru is reconstructed Norse paganism. It comes in generally two strains: a peaceful, non-racist form that most followers embrace, and a violent Neo-Nazi form. Confusing the two is a major Berserk Button.
- Chaotic paganism is a left hand path that has some actual similarities to Satanism below in love for embracing contradiction and theatrics and for embracing or reimaging of traditionally "evil" or "negative" concepts, deities, energies, etcetera. That said very importantly it is not Satanism, and is a solitary pagan spiritual path rather than one affiliated with any church or religion - and which is actually most similar to neo-shamanism below as a "kitchen sink" or "cafeteria" path. There are some texts, but any specific practitioner is free to reject them and create their own way.
- Druidry is a reconstruction of Celtic paganism.
- Kemetic belief is a reconstruction of Egyptian paganism.
- Neo-shamanism is (sometimes unfairly seen as) the New Age Retro Hippie / Granola Girl "kitchen sink" neopagan path, as it is about sampling from different beliefs and creating one's own practice, but generally avoiding "negative" deities, energies, and spirits and being more about healing and lightwork. In it, too, there are some texts, but any specific person is free to reject them and create their own way, and most remain solitary, though some will affiliate with Buddhism or Wicca or, in a more appropriating way, Native American spirituality below.
- While Wicca is the predominant neopagan faith, there are plenty of others, though they have even smaller representations:
- There are about 650,000 Sikhs in the United States, most of them living in California. Despite their own history of clashing with Muslims, Sikhs wound up getting caught up in the wave of Islamophobia that occurred after 9/11, due to the fact that many of them wear turbans. The first Indian (or indeed Asian of any kind) to serve in Congress was Dalip Singh Saund, a Sikh from California who served from 1957 to 1963.
- Native American spirituality has still held on at reservations.
- Scientology is a... highly controversial religious movement founded in the United States. They claim to have over three million members in America, but most estimates put their numbers at below 100,000. They have lots of missionaries giving "Free Stress Tests" or the like, and many are attracted from the "Free" offer. However, most "converts" leave soon after they find out about the exorbitant prices one must pay to continue on the Bridge, as well as the practice of shunning non-members or "SPs". They are extremely well-represented in Hollywood, due to their practice of proselytizing toward celebrities (who are usually rich enough to afford the thousands of dollars that their programs cost) in order to use them as spokespeople to gain further converts. They also have a strong presence in Clearwater, Florida, often called "Scientology's Town", where they have their headquarters.
- Baha'i is a monotheistic religion with millions of followers around the world. Baha'i in America are divided among Persians, many of whom fled the Islamic Revolution in 1979 (which, to say the least, did not like them) and the sort of New Age Retro Hippie-types who might have become Buddhist but preferred something more Abrahamic. The most notable Baha'i in America is none other than Rainn Wilson, AKA Dwight Schute, whose parents were of the second category and raised him in the faith while living in a houseboat off the coast of Washington State.
- The Church of Satan was founded in The Sixties by Anton LaVey in San Francisco. Sources state that there are about 10-20,000 official members of the Church of Satan in the United States, and there are possibly tens of thousands more who adhere to the philosophy or one of its offshoots. Despite their name and reputation, they do not actually worship Satan, being an atheistic organization rooted in a mix of pseudo-Nietzschean philosophy and the theatrics of Aleister Crowley and other occultists. They chose the name because they feel that Satan, the original rebel in Christian theology, is a role model for people to look up to, and that the Christian message of tolerance, humility, and egalitarianism is self-destructive for both individuals and society. (That, and nothing draws attention like calling yourself the Church of Satan.) Note that the Church of Satan is not to be confused with...
- The Worldwide Satanic Church of Evil, which is not a real church -- although that hasn't stopped thousands of Urban Legends. Since time immemorial, many religious groups have claimed that there is an evil Cult that performs occult rituals, Human Sacrifice, and other evil acts.  Fear of Satanism turned into a moral panic back in The Eighties following the publication of Michelle Remembers, a book purporting to be an expose of an underground, worldwide Satanic organization with millions of members performing horrific acts on children. The ensuing panic over "Satanic ritual abuse" did lasting damage to the day care industry (which was hit hard by dozens of allegations of Satanic abuse) and social services (which jumped onto the Satanism bandwagon early, and saw a huge backlash once Satanic abuse became discredited), and even saw Proctor & Gamble forced to change its logo following accusations that its original logo was Satanic (they would be awarded $19 million in damages from the people who spread the rumors, which had caused their stock to plummet).
- And even then, this was a huge deal at the time; many Protestants were still concerned that Kennedy would be a puppet of Rome. He had to give a speech specifically affirming that his first loyalty was to the US Constitution, and that his actions as President would not, and should not, be bound by the dictates of the Church. This speech wound up becoming a major landmark in the aforementioned separation of church and state.
- due to large scale European immigration in the later part of the 19th century in the former, and the historic French influence in the latter
- not to be confused with the Presbyterian Church (U.S.A.), which is mainline Protestant
- As always, there are exceptions to every rule. A number of more liberal, mainline churches call themselves evangelical (such as the Evangelical Lutheran Church of America), and a substantial minority of individual evangelicals, particularly younger ones, reject Biblical literalism.
- These churches bring in millions of dollars annually, tax free, and their leaders also get tax breaks. It has caused some friction. Think "money changers in the temple".
- The two are separate because the Eastern Orthodox Church accepts all seven ecumenical councils while the Oriental Orthodox only accept the first three; they're known as the Non-Chalcedonian churches because they rejected the Council of Chalcedon.
- Mostly in New York, New Jersey, California, and Michigan. If the last one sounds weird, Michigan has a large population of Arabic-speakers -- for the Egyptian Copts -- and Armenians.
- Most notably, in the '08 primary Mike Huckabee, a competing candidate who was running as a social conservative, caused controversy after making some derisive comments about Romney's Mormonism.
- They were heavily involved in the Civil Rights Movement, and it was a Jewish woman who wrote The Feminine Mystique, the book often credited with kick-starting the second wave of feminism.
- Interestingly, before the attacks, most Muslims voted Republican due to their affluence and social conservatism.
- The title of "least religious state" often varies between Vermont, Colorado, Oregon and Washington depending on the year and the survey.
- An example of the type of treatment that atheists get in America -- former President George H. W. Bush once stated in an interview on the campaign trail that atheists couldn't be considered citizens, as America is "one nation under God."
- For the record, the reason there aren't more Buddhist congresspeople despite the relatively large number of Asian representatives is that most Asian American politicians are Christians.
- LaVey claimed inspiration from Objectivism, but his philosophy is significantly different.
- The Church of Satan actually did perform Black Masses during their heyday, but they were purely for show and to spark controversy.