Great question. Here's a very long cut-and-paste from the Carnegie-Mellon University Newman Club that describes them all, and all are ultimately under the Pope in Rome:
Western Rites and Church
ROMAN CHURCH
The Church of Rome is the Primal See of the world and the Patriarchal See of Western Christianity. Founded by St. Peter in 42 it was consecrated by the blood of Sts. Peter and Paul during the persecution of Nero (63-67 AD). It has maintained a continual existence since then and is the source of a family of Rites in the West. While the origin of the current Rite, even in the reform of Vatican II, can be directly traced to only the 4th century, these connections point to an ancient apostolic tradition brought to that city that was decidedly Jewish in origin.
After the Council of Trent (1560) it was necessary to consolidate liturgical doctrine and practice in the face of the Protestant Revolt. Thus, Pope St. Pius V imposed the Rite of Rome on the Latin Church (the Rites subject to him in his capacity as Patriarch of the West), allowing only smaller Western Rites with hundreds of years of history to remain. Many younger Rites of particular dioceses or regions ceased to exist. So, the term "Roman" Rite wasn?t created until the 1500s.
Latin
* Rite of an overwhelming majority of Roman Catholics, and majority of Catholics in general.
* Named because of the use of Latin in the Liturgy, and is still part of Canon Law: "The eucharistic celebration is to be carried out either in the Latin language or in another language, provided the liturgical texts have been lawfully approved." (Cannon 928, 1983 Code)
Ambrosian
* The Rite of the Archdiocese of Milan, Italy
* Thought to be of an early origin; probably consolidated, but not originated, by St. Ambrose in the 4th century.
Bragan
* Rite of the Archdiocese of Braga, the Primal See of Portugal, it is only occasionally used.
* Derives from the 12th century or earlier.
Mozarabic
* Confined to the Rite of the Iberian peninsula-specifically, the Cathedral of the Archdiocese of Toledo, Spain, and 6 of its parishes.
* Known to exist from at least the 6th century, but probably with roots to the original evangelization.
* Beginning in the 11th century it was generally replaced by the Roman Rite, although it has remained in the locations listed above. Its celebration today is generally semi-private.
Dominican
* Rite of the Order of Friars Preacher (OP), founded by St. Dominic in 1215.
Carmelite
* Rite of the Order of Carmel, whose modern foundation was by St. Berthold c.1154.
Carthusian
* Rite of the Cathusian Order founded by St. Bruno in 1084.
Eastern Rites and Churches
They have their own hierarchy distinct from the Latin Rite, system of governance (synods) and general law (the Code of Canons for the Eastern Churches). The Supreme Pontiff exercises his authority over them through the Congregation for the Eastern Churches.
ANTIOCHIAN CHURCH
The Church of Antioch in Syria (on the Mediterranean coast) is considered an apostolic see by virtue of having been founded by St. Peter. It was one of the ancient centers of the Church, as the New Testament attests, and is the source of a family of similar Rites using the ancient Syriac language (the Semitic dialect used in Jesus' time and better known as Aramaic). Its Liturgy is attributed to St. James and the Church of Jerusalem. It is divided up into two main areas: the Eastern Syrian Church, and the Western Syrian Church.
1. EAST SYRIAN
Chaldean
* Following Arian heresy in 4th century (Christ was a perfect creature, but not Divine), Nestorius of Constantinople taught that Christ was Divine and human?because Christ was "2 persons".
* As the Council of Nicea (431) corrected this error, Nestorious and his followers fled to Persia and other parts of the world
* Some gradually (and slowly) returned to the Church in 1692.
Syro-Malabarese
* Part of the defectors that followed Nestorius, they settled in South India, and continued their use of the East Syriac liturgy.
* With the help of Jesuit missionaries, they returned to communion with Rome in the 16th century.
2. WEST SYRIAN
Maronite
* Syrian Christians who retired into the mountains of Lebanon for protection against political and religious enemies.
* When the Crusaders ?ran into them? in the 12th century, they immediately proclaimed their unity with the Catholic Church, and they still pride themselves to this day that they have never formally separated themselves from it.
* Because they have never left communion with Rome, they are one of the few Rites that has no non-Catholic counterpart, e.g. there are no Orthodox or Monophysite Maronites--the only Maronites are Catholics.
Syriac
* 20 years after Nestorius? preached his heresy, the Monophysite heresy taught that Christ was indeed 1 person--with just one nature (Divine). (Monophysite means "one-natured)
* The Council of Chalcedon (451) corrected this, but many Syrians and Egyptians followed this teaching out of the Church (many in Egypt still hold this view to this day).
* They eventually grew into the Pure Syrian Rite, and many returned to Rome in 1781.
Malankarese
* Another group of Catholics in South India, they were originally evangelized by St. Thomas but later left the Church during a heresy.
* Eventually reunited with Rome in 1930 under the guidance of their charismatic leader Mar Ivanios.
ALEXANDRIAN CHURCH
The Church of Alexandria in Egypt was one of the original centers of Christianity, since like Rome and Antioch it had a large Jewish population, which was the initial object of apostolic evangelization. Its Liturgy is attributed to St. Mark the evangelist, and shows the later influence of the Byzantine Liturgy, in addition to its unique elements.
Coptic
* Due to their dependence on Egypt, many Coptic Catholics in Ethiopia/other parts of Africa followed the Monophysite heresy in the 5th.
* A small minority of Coptics returned to the fullness of the Faith in 1741, and retained the customs they had for the prior 12 centuries.
* A majority of Copts/Coptics in the world are not Catholics, and elect their own "pope" in Alexandria. They are often lumped together with the Orthodox Church (much to the Orthodox?s dismay, since they also view them as "heretics").
Ethiopian/Abyssinian
* Coptic Christians in Ethiopia who returned to Rome in 1846.
BYZANTINE CHURCH
The Church of Constantinople became the political and religious center of the eastern Roman Empire after the Emperor Constantine built a new capital there (324-330) on the site of the ancient town of Byzantium. Constantinople developed its own liturgical rite from the Liturgy of St. James, in one form as modified by St. Basil, and in a more commonly used form, as modified by St. John Chrysostom. After 1054, except for brief periods of reunion, most Byzantine Christians have not been in communion with Rome. They make up the Orthodox Churches of the East, whose titular head is the Patriarch of Constantinople. The Orthodox Churches are mostly self-headed, united to each other by communion with Constantinople, which exercises no real authority over them. They are typically divided into Churches along national lines (such as the Russian Orthodox Church, Greek Orthodox Church, etc.). Those that have returned to communion with the Holy See are represented among the Eastern Churches and Eastern Rites of the Catholic Church. The two main Churches in the Byzantine Catholic Church are the Armenian and the Byzantine.
1. ARMENIAN
* It is considered either its own Rite or an older version of the Byzantine, although its exact liturgical form is not used by any other Byzantine Rite.
* Composed of Catholics from the first people to convert as a nation, the Armenian Rite was developed under the guidance of St. Gregory the Illuminator before the Great councils in the 4th century.
* The Armenians sided with the Monophysites and left the Church in 451, but many returned during the Crusades. Most of the Armenians in the world are members of the Orthodox Church and are not in union with Rome.
2. BYZANTINE
Even though the Polish, Czech, Slovenian, most of Slovak and Croatian people now belong to the Roman Rite, originally they did not and were ?pushed? under the authority of German bishops as the Carolingian military began to subdue these areas. The majority of those who practice the Byzantine Rite (often popularly called "Greek Catholics" or "Greek Rite") are not, in fact, Greek at all, but Slavs. The Slavic language is still maintained in use within the liturgy because it is still a 'vernacular' language within the liturgical meaning of the term as people still speak it in their homes. The Romanians and Hungarians do not use Slavonic, however, but their own languages, and those of Italy and Sicily use Greek rather than Slavonic, Italian or Latin. The use of English is widespread in the United States in all usages of the Byzantine Rite.
Albanian
* Albanian Christians, numbering only 1400 today, who resumed communion with Rome in 1628.
Belarussian/Byelorussian
* Unknown number of Belarussians who returned to Rome in the 17th century.
Bulgarian
* Some of the Bulgarian Christians left the Orthodox Church and returned to Rome in 1861.
Czech
* The Czech Rite was recently organized into a jurisdiction in 1996.
Krizevci
* Croatian Catholics of Byzantine Rite who resumed communion with Rome in 1611. Most Croatians are Roman Rite.
Greek
* Comprised of Greek Christians who returned to Rome in 1829, there are only 2,500 Greek Catholics in Greece, Asia Minor (Turkey) and Europe.
Hungarian
* Descendants of Ruthenians who returned to Rome in 1646.
Italo-Albanian
* Never separated from Rome, these 60,000 Byzantine Rite Catholics are found in Italy, Sicily and the Americas.
Melkite
* Catholics from among those separated from Rome in Syria and Egypt who resumed Communion with Rome at the time of the Crusades, although definitive union only came in the 18th century.
* "Melk" in Syriac means "king", and they were known as "the King?s Men" because the minority of Melkites in Egypt frequently turned to the Emperor in Constantinople for assistance during their persecution.
* Their customs gradually slipped into disuse over time, and eventually adopted the Byzantium customs held to this day.
Romanian
* Romanians who returned to Rome in 1697, most Romanian Christians are Romanian Orthodox.
Russian
* Russians who returned to communion with Rome in 1905; the vast majority of Christians in Russia are Russian Orthodox.
Ruthenian
* Catholics from among those separated from Rome in Russia, Hungary and Croatia who reunited with Rome in 1596 (Brest-Litovsk) and 1646 (Uzhorod).
Slovak
* Byzantine Rite Catholics of Slovakian origin numbering 225,000 and found in Slovakia and Canada.
Ukrainian
* Catholics from among those separated from Rome by the Greek Schism and reunited about 1595.
* During the Soviet era, Ukrainian Catholics were violently forced to join the Ukrainian Orthodox Church. Their hierarchy, which continued to exist outside of Russia, has since be re-established in the Ukraine.