We may know more isolated facts about the Whoniverse. We may be more emotionally moved by the Buffy Verse, but in terms of a coherent sense of history, cross-series and cross-media continuity, you really can't beat the Trek Verse.
The Trek Verse was invented by Gene Roddenberry for Star Trek the Original Series. Initially no more detailed or self-consistent than any other Speculative Fiction 'verse, it has become a place with a continuity which outstrips that of any other project on television. It is built up by the Canon details from five television series and ten feature films; there is also a myriad of original novels and an Animated Adaptation that are not officially canon according to Paramount Pictures, which owns the franchise, but many fans consider them so anyway. (As for the several Comic Book Adaptations, their quality varies wildly, so at least some of the older ones have fallen prey to Fanon Discontinuity.) It also spawns vast amounts of Fanon, in the form of more Fanfic than the human mind can safely comprehend.
At least one planned series (Assignment: Earth) was to be set in the Trek Verse during the 1960s, but did not come to pass.
The Trek Verse diverges from our universe some time between 1965 and 2000. The major divergence arguably occurred some time in the 1960s, with the creation of a group of genetically enhanced humans. The differences did not become visible until the 1990s, though, when that group (which included Khan), now adults, launched an international conflict called the Eugenics Wars. The exact impact of this war is hard to deduce, as there have been several retcons applied to allow Time Travel stories to a recognizable contemporary Earth. At one point, it appeared that the Eugenics Wars had been retconned into the late 21st century, but that retcon was itself retconned into a simple inaccuracy (A character saying "two hundred years" when "three hundred years" was called for).
The "active" history of the Trek Verse covers the period of time during which mankind has spread out into nearby space, a period of approximately 300 years, separated into three eras:
- The Enterprise era: Mid-22nd century. Man's first forays into deep space. This era culminates with the foundation of a United Federation of Planets, of which Earth is a signatory member. (ENT)
- The Classic era: Late 23rd century. A period of rapid expansion for the Federation. It is before or during this period that man reaches and surpasses the technological level of the rest of the major local species and becomes the dominant player in local politics. (TOS, TAS)
- The Modern era: Mid to late 24th century. A period of internal conflict and consolidation. Despite tacit peace with the other local major powers, wars with distant powers such as the Borg and the Dominion weaken the Federation. (TNG, DS9, VOY)
It also includes at least three separate continuities -- the main 'verse, the mirror 'verse (of the Terran Empire and later, uglier events in which the former subjects of the Empire are subject to backlash by everyone else), and the JJ Abrams movie continuity, with hints of others, the most likely candidate being the alternate future of Enterprise's Temporal Cold War.   It's almost complex as the many continuities of Transformers, but at least the Trek Verse mercifully interconnects them in ways that mostly make sense.
- Humans: A race that is characterized, ironically, by their lack of strong defining characteristics, as well as their disproportionately large numbers within the Federation colonies and Starfleet, especially during the classic era. Other unique aspects of humans include the comparatively high level of diversity within their own species, and their familiarity with their planet's pre-apocalyptic history and culture; which was a time of rampant violence, greed, and ignorance before the Third World War ushered in a new cultural era. Earth, the human homeworld, is home to the Federation's governmental bodies and Starfleet, its military force; and is located in Sector 001 on Federation maps.
- Vulcans: A logical, long-lived species (who suppress their emotions) with close ties to humanity. They are best recognized for their pointed ears, Moe Howard haircuts, a distinctive salute (see Cross-Cultural Handshake), and a clever defensive attack wherein they can render a person unconscious by squeezing the neck (the famed Vulcan Nerve Pinch). Like humans, Vulcans went through a cultural turning point (although theirs was much more drastic), and their intense devotion to logic is a backlash against their wild, vicious past. Much is made of the contrast between the rational, collected Vulcans and the more emotional and impulsive human characters.
- Bajorans: A mostly peaceful and spiritual race hardened by the rape of their homeworld by the Cardassians, they play a large role in Deep Space Nine, in which the titular space station hovers over their planet. The Federation wants them to become a member state but the two are opposed in some ways, including the Bajoran religion, especially the slightly xenophobic hardliner faction that wants both the Cardassians and the Federation to stay out.
- Klingons: a Proud Warrior Race of Rubber Forehead Aliens (originally white guys wearing fu manchus and shoe polish in TOS; retconned in Star Trek: Enterprise), started out as enemies of the Federation, but later became tacit allies. Later stories focus on the internal struggles of the Klingon Empire as a power on the wane.
- Romulans: Secretive diaspora Vulcan sub-species who rejected the Vulcan drive for logic, but have cloaking devices, which makes up for it. Highly xenophobic and reliant on their secret police to keep order, they are loosely inspired by the Roman Empire and their territory is the Romulan Star Empire. They are also somewhat a Proud Warrior Race but have fewer qualms about sneaky tactics and value honor for the Empire higher than personal honor. They once got along with the Klingons but these fundamental differences and some nasty wars mean Romulans and Klingons hate each other's guts. The 2009 Trek reboot movie makes it canon that the Romulan homeworld was destroyed by a supernova, sending Nero back in time for revenge and starting the Alternate Timeline.
- Ferengi: An entire race of slimy capitalists. The big-eared, big-nosed Ferengi value profit above all else, cheat and swindle as a way of life, have big ears and noses, and are played by primarily Jewish actors. The Ferengi were originally meant to be the new Big Bad for The Next Generation in the way that the Klingons were for the original series, but that fell through after one or two episodes when it became apparent how silly they were. Ferengi were later relegated to comedy relief, most famously in the character Quark on Deep Space Nine.
- Cardassians: Even more Scary Dogmatic Aliens than the Romulans, introduced in the Next Generation era and a main antagonist of Deep Space Nine; Expansionist, collectivist and many Nineteen Eighty-Four overtones; they are brutal as colonial overlords and merciless with dissenters. They occupied the Bajoran homeworld until La Résistance got too much for them and actively try to subvert the treaty that forms the Cardassian Neutral Zone in order to control it. They later side with The Dominion against the Federation. This ends horrifically for them, and by the latest point in the official timeline they've practically been reduced to a rump state due to the complete devastation of their entire nation, both homeworld and colonies.
- The Borg: Cybernetic collective skilled in adaptation and assimilation. Took over the role of thinly veiled stand-ins for communists from the Klingons in the Modern era. Generally treated as the single largest threat to the Federation, in part because they more or less cannot be negotiated with, have access to the knowledge of thousands of species and trillions of individuals and can adapt very quickly to almost any weapon used against them, rendering it useless, and partly because they are in some ways the Federation's dark reflection. A group that goes around incorporating other cultures into their own, in order to make the whole stronger, more complete, and more "perfect"? Did the writer just describe the Federation... or the Borg?
- The Dominion: The Empire from another part of space, set to take over our side of the galaxy. Much of Star Trek: Deep Space Nine revolves around the war with them, and the consequences of this Arc are sometimes felt in the other series. They also get to be stand-ins for the evil of drugs, as their warriors are addicted to one to keep them in line.
The Milky Way galaxy is divided into four 90-degree wedges called quadrants. These are:
- The Alpha Quadrant. This is home to the Humans, Vulcans, and a number of other minor races, all of whom have banded together to form The United Federation of Planets. It's also home to the Bajorans, Ferengi, and Cardassians. Portions of the Klingon and Romulan Empires also extend into the Alpha Quadrant.
- The Federation occupies a region of space roughly 8000 light-years across. Since it's 50,000 light-years from the center of the galaxy to the edge, the Federation covers less than 3% of the alpha quadrant.
- Earth is located on the border of the Alpha and Beta Quadrants.
- The Beta Quadrant. Major parts of the Klingon and Romulan empires are here. The Federation, likewise, claims some territories inside the Beta Quadrant. This is the least-discussed quadrant in the various TV series and movies, and as such little is known about it.
- The Gamma Quadrant. Home to the Dominion. A permanent wormhole connects the Gamma Quadrant to the star system where the Bajorans live.
- The Delta Quadrant. Home to the Borg, and several other species. This is the quadrant that the U.S.S. Voyager was lost in for 7 years.
Technology in the Star Trek universe is as varied as the scriptwriters who invented it. However, despite some outright discontinuities in the availability or even existence of various pieces of Treknology, there are certain staples that show up again and again as the standard means by which the Federation (and most of the other major players) get things done:
- Faster-Than-Light Travel, in the form of Warp Drive. Starships can travel several times, or even several hundred times, the speed of light -- but only under rare, non-repeatable conditions can they cross the entire galaxy in less than several decades. Starships under warp are fully aware of the sub-light universe they're travelling through, and may even (Depending on the Writer) engage each other in combat while travelling at warp speed.
- FTL Radio, over subspace. Depending on the Writer, subspace radio either propagates infinitely fast (allowing for instantaneous two-way communication between distant starships and Starfleet HQ), or takes days to cross the larger interstellar distances. A quadrant-spanning subspace relay network is usually involved, whether deliberately mentioned or not.
- Deflector Shields. These are standard equipment on most starships. They consume a significant (but not excessive) amount of power, can withstand a sizable amount of punishment, and are weakened in precisely predictable ways when damaged ("Shields down to 40%, Captain!"). Scaled-down versions of this technology are used to create smaller force fields, for such uses as prison doors that are vulnerable to power failures.
- Transporters and Teleporters, called simply Transporters. They can operate over distances of several thousand kilometers, but are not long ranged enough to beam a person from the Earth to the Moon (unless someone comes up with a Techno Babble way of doing so that only works for that one episode). They cannot operate through a starship's Deflector Shields. Several other local conditions can interfere with the operation of Transporters, usually in direct proportion to how much immediate danger an away-team is in.
- Ray Gun, in the form of Federation phasers and Klingon disruptors. There are small handheld, and large starship-mounted, varieties. Hand-held phasers have a "stun" setting, useful for knocking out an adversary without killing it. Ship-mounted varieties are usually powerful enough to weaken an enemy ship's shields, but not so powerful as to destroy their target in a single hit. Depending on the scriptwriter (and the era), phaser beams travel either at the speed of light, or many times faster than light.
- Tractor Beam. Sometimes visible, sometimes not. Fighting a tractor beam with your ship's engines can put a heavy strain on your hull, and may threaten to tear you apart.
- Everything Sensor, usually referred to as simply the ship's sensors. They can detect life forms, time travel, the entire subatomic particle zoo, and even alterations to the fabric of the universe, and can pinpoint not only the direction to any of these oddities but also the distance. However, their ability to locate a crew member planetside seems to be inversely proportional to the amount of peril said crew member finds himself in.
- Stealth in Space, in the form of the Cloaking Device. Thanks to the Treaty of Algeron, only the Romulans and Klingons posess this technology ... at least, officially.
- Artificial Gravity. Ubiquitous on all starships and space stations, it is the most robust piece of technology in the 'verse -- it has never failed in battle and has only been deliberately deactivated a handful of times.
Starting in the Next Generation era, we also got:
- Hard Light, used for entertainment on the Holodeck. Some Holodeck programs can take on a mind of their own, though the computer that controls them never does. The system can also be used to create artificial crew members wherever holographic emitters have been installed, including emergency medical doctors.