Orthodox Christianity
One of the two Churches resulting from the 11th century Great Schism of the Christian Church, the other being the Roman Catholic Church. The schism gradually developed; for several centuries before the formal break, contact had been intermittent between the Eastern and Western churches, with each pretty much running its own show. As a result, when the schism occurred, each side could honestly believe "we never changed, they left us". The "Great Schism" is generally dated from when a Papal legate and the Patriarch of Constantinople excommunicated each other in 1054, but it had been a long time coming; go to the Other Wiki if you want all the gory details.
No Pope (one of the main reasons for the split, and the biggest obstacle to reunification in the present); the church is instead led by several Patriarchs, each responsible for a different region. Cyril I is the current Patriarch of the Russian Orthodox Church, by far the largest church, but the Ecumenical Patriarch of Constantinople (currently Bartholomew I) is considered "first among equals".
Married men can become priests, priests' wives are important in the church. Some jurisdictions follow a different calendar from the Catholic church (the Julian calendar, originated in Ancient Rome, which is considered the "holy calendar"), thus Christmas takes place January 7th. Others use a Revised Julian calendar that is almost exactly the same as the secular calendar (but will diverge at some point in the future); however, almost all of those churches still calculate Easter and related holidays according to the old calendar (yes, it's complicated). Easter ("Paskha") can be up to several weeks after Catholic Easter, both from the difference in calendar and by being determined by a different method.
The look-and-feel of the religion is very similar to Catholicism, much more than to Protestantism: there are bishops, monks, nuns, saints and other pre-Reformational trappings. Church services have lots of chanting, gold vestments, incense, candles. But the liturgical languages are Greek, Old Church Slavonic or the national language of the country the particular Church is from (e.g., Japanese in Japan, English in the OCA, etc.) and the spirituality is decidedly more "Eastern" than western churches (see http://en.wikipedia.org/wiki/Hesychasm for an example). Orthodox Christians make the sign of the cross up-down-right-left, whereas Catholics go up-down-left-right; in some spy stories, a westerner might give himself away by crossing himself the wrong way while saying grace.
This religion has strong historical ties to the Eastern Roman Empire, which is why a good deal of the religion is centred on historically Greek areas (Alexandria, in Egypt, for instance), and beyond Russia the religion is predominant in much of the Balkans including Greece.
There are also "Eastern Catholic" churches, which look and smell like Orthodox churches, and have many Orthodox practices (including even married priests), but are in communion with the Roman Catholic Church and accept the Pope's authority. The Ukrainian Catholic Church is one of the largest and best known of these.
Note about married priests: a married man may become a priest, but a priest may not marry. Thus, while a married man can continue to live with and have marital relations with his wife (subject to fasting restrictions), he cannot take a wife after he has been ordained, regardless of whether or not he was married when ordained. The same rules apply to Catholic deacons and the rare instances of married Latin Rite Catholic priests.
Oriental Orthodoxy
A communion of Churches who consider themselves Orthodox and differ from the Eastern Orthodox because of their Christology (i.e. their beliefs about the nature of Jesus Christ). They are also known as the non-Chalcedonian churches, as they reject the Christological dogma promulgated by the Council of Chalcedon (about which see more below). These churches are the Coptic (i.e. Egyptian), Armenian, Syriac, Ethiopian, Eritrean and Indian churches. The largest of these is the Ethiopian Orthodox Tewahedo Church, currently headed by Patriarch Abuna Paulos. Much as the Eastern Orthodox churches are in full communion with each other and recognize the Patriarch of Constantinople as first among equals, the Oriental churches are also in full communion with each other and recognize the Patriarch of Alexandria (who resides in Cairo these days, confusingly enough) as first among equals. This is despite the fact that the Patriarch of Alexandria's title is "Pope," but he doesn't seem to mind. The post of Coptic Pope of Alexandria is currently vacant; the vacancy is expected to persist for at least several months, as the selection process involves a series of consultations among Coptic clergy and laity to narrow the field down to three candidates, after which the Pope is chosen by literally having a blindfolded child pull one of the three's name out of a hat (or some other vessel). The most recent Pope, Shenouda III, died in March 2012. He had gained accolades for his campaigns for Christian unity but also caught a bit of flack (though not too much) for backing the wrong side in the early days of the Egyptian Revolution of 2011 (it was understandable--it wasn't immediately clear that the rights of Christians would be respected; Coptic revolutionaries were nevertheless understandably pissed off at their Pope for a while).
Their Christology, referred to as miaphysitism, says that Jesus Christ has one incarnate nature that is both human and divine. The Chalcedonian formula is that Jesus had a human nature and a divine nature in hypostatic union. When the split occurred, Christology was Serious Business (people were occasionally killed in the schism). Today, the differences (particularly in the face of Protestantism) appear rather picayune (Larry Gonick joked that the dispute between the Eastern and Oriental amounted to one saying "It says in [such and such bit of scripture] that the Lord's Enemy--namely YOU--smells like rotten eggs!" and the other saying, "Strangely, I agree verbatim..."). Recently leaders of the two churches have stated the both definitions are merely two different ways of saying the same thing and there is hope that there will be reconciliation.