Some years ago I taught a course at Milwaukee's Lafarge Institute entitled "Introduction to the Eastern Churches." It turned out to be a popular course. And the running joke was that the student needed a "score-card," in order to understand the terminology: Eastern Orthodox, Coptic, Chaldeans, Maronites, Byzantine-Catholics, and on and on.
From the perspective of world-wide Christianity, all the Eastern Christians put together are numerically small. From an historical perspective, small as they may be, they are very significant in that they represent some of the earliest continuously existing Christian communities. In the Acts of the Apostles it states matter- of-factly that "it was at Antioch that the disciples were first called Christians." Today there are no less than 5 ancient communities who trace their origin to the (once) great city of Antioch.
Early on in the development of the organization of the Church throughout the world it was agreed upon that the bishops of certain major cities of the Empire enjoyed preeminence among the bishops. These were the bishops of Rome, Alexandria, Antioch and Jerusalem. Later Constantinople was added immediately after Rome, being as it was, the "New Rome." This caused a problem because Rome and Alexandria both resented this "intrusion" in the ancient order of things.
At any rate, situations such as this caused rifts great and small amongst the various Churches. Theological differences, now considered semantic rather than dogmatic, caused further separations. The fall of the Empire in the West caused a whole new world-view to evolve there - including a brand-new Roman Empire: that of the formerly barbarian kings. Remember that in the order of the chief bishops or patriarchs, there came to be five - with only one in the West.
The relationship of the Bishop of Rome to the other Bishops of the other Apostolic Churches was a matter of discussion, opinion, and even doctrine over the centuries. Constantinople and its Church flourished and spread throughout the Balkans and the lands of the Eastern Slavs (Ukraine, Russia, Belarus) with only occasional contact with "Old Rome." The Crusades were the occasion of some contacts, which did not endear the Western Christians to their Eastern brethren. (To be fair, there were events which rarely are mentioned, such as the slaughter of the Latin Christians of Constantinople)
Now, I have admittedly not expressed all of the above in a scholarly way, but it brings me to a point: By the time of Constantinople's fall to the Muslims in the Fifteenth Century, ancient, apostolic, catholic, orthodox Christianity had become very much divided. The Copts of Egypt, the Ethiopians and the Armenians maintained Communion with each other. The Antiochenes were divided between the "Melkites," i.e. Byzantines and the Syriacs. There were Nestorian communities from Baghdad to China. There were ancient communities in India. And there was Rome and Western Europe.
Over the years there were attempts at reunion. Notable among these was the Council of Florence (in the 1430's) which actually managed to unite all the Eastern Churches with Rome. Though the bishops managed to come to tenuous agreements, when they returned home to Greece, Russia, Egypt, etc. the "union" fell apart. It simply was not accepted.
[Interesting note: there were Eastern Christians in southern Italy who had never been separated from union with Rome. Also, the Maronite Christians of Lebanon have claimed that their Church also never separated from Communion with Rome.]
A century or so after the failed union of Florence, the popes sought in various ways to reach out to the Eastern Churches hoping for another reunion. What happened -briefly- is that communities of Eastern Christians entered into union with the Roman Catholic Church. Primary among these were the Eastern Slavs - called collectively the "Rus" or "Rusyns." These were Orthodox Christians living in the general area of Ukraine and Belarus. The union took place in 1595-96. Most of the bishops, with their Metropolitan Archbishop of Kiev, agreed to articles which would preserve their legitimate, ancient, Byzantine traditions of worship, government, etc. while recognizing the authority of the Pope of Rome. This "program" was called the Unia - a Polish term which means "union." The Christians of the Kievan Metropolia were virtually all within the government of the Polish-Lithuanian Commonwealth and there's no doubt that political considerations played an important role in the "Unia." Though related ethnically to the Russians (Muscovites), there had been little contact between the Church of Moscow and that of Kiev. In fact, the Metropolitans of Kiev were confirmed directly by the Patriarch of Constantinople.
Those who entered into union with Rome came to be called Uniates. This term has become more and more a pejorative. Personally, I like the term. When I was a youngster already interested in Eastern Christianity, it was a pretty standard term. Plus, it means what it says: United. But no one else seems to like it. Instead, the descendants of these original Uniates are called Byzantine Catholics. Or Greek Catholics - though none of them are Greek. Those who are not of Byzantine tradition are called Eastern Catholics. More often they are called by their particular names: Chaldean Catholics, Malabarese Catholics, Coptic Catholics, etc. Nowadays there is much hyphenation, as in "Ukrainian-Greek-Catholic" or "Melkite-Greek-Catholic." Not too long ago they were collectively called Eastern Rite Catholics. This was inadequate because it implied that the Eastern Catholics were "regular" Catholics except that their "rite" or ritual was exotic.
The preferred term in many circles is "Orthodox in union with Rome." This reflects a newer self-understanding: that we Eastern Catholics are Orthodox Christians (just like the Russian Orthodox and the Serbian Orthodox), except that we have entered into union with Rome. Eminent Russian philosopher Vladimir Soloviev would agree with this term. He converted to Catholicism based on his conclusion that to be fully Orthodox one had to be in union with the Roman Church. Of course, for most people "Orthodox in union with Rome" is a kind of oxymoron, because the "Orthodox" are those who are not in union with Rome.
What I find interesting is that, in the Polish language, whence the term "uniate" is derived, the Eastern Catholics were referred to as "unici prawoslawni" which means, simply enough "United Orthodox," or, "Orthodox in Union."
In the long run, it doesn't do much to resolve the need for a "score-card." And there is little danger of anyone finding a sign that reads "St. Thomas Uniate Church." It is sad that history has heaped up division after division, false unions, partial unions, forced unions. Uniates - oops! - I mean Eastern Catholics still are in need of a self-understanding that expresses the reality of their union, challenges any imposed or embraced character of second-class status - and whatever resentments are bound to attach to such a status, renew the beautiful and valuable treasures of their own spiritual and liturgical patrimony, without apologizing for their very existence - either to the West or the East.