The Eastern Rites are the Christian churches of Eastern Europe and the Middle East that are in full communion with the Roman Catholic Church but do not follow the Latin Rite. They are also called Eastern Catholic or Uniate churches.
Western (or "Latin-Rite") Catholic bishops are subject directly to the Pope, but each Eastern-rite Catholic bishop is subject indirectly to the pope via one of six Catholic "patriarchs of the east", who sit in Alexandria, Antioch, Antelias, Baghdad, Beirut, and Damascus. (There is a seventh patriarch of the east in Jerusalem, but his church follows the Latin Rite.)
These churches accept Catholic dogma, but retain their own hierarchies and liturgies, and follow some laws and customs that differ from those of Western church. For example, their priests need not be celibate, and their parish priests, rather than diocesan bishops, normally confirm parishioners, using the chrismation rite rather than the rite used in the west.
It should be noted that not all the "Western" churches follow the Roman rite, in fact, several Italian dioceses have their own rites and usually the orders will have their own rites (Dominicans, Jesuits).
List of Eastern-Rite churches