A rite is sort of like a family of churches within the family of the whole Catholic Church. In the early Church, there were five sees based in the most significant cities in the ancient world, each with apostolic origin:
Jerusalem, Antioch, Alexandria, Rome, Constantinople
Today, there are a total of 24 self-governing sui juris churches that make up the Universal Catholic Church as a whole. These churches fall into the six different rites:
Latin, or Roman Rite
West Syriac (Maronite and Syro-Malankar)
East Syriac (Chaldean and Syro-Malabar)
Alexandrian (Coptic and Ethiopian).
The Byzantine, Armenian, and Syriac rites all grew out of the See of Antioch.
The most common Byzantine Catholic churches in the United States are Byzantine Ruthenian, Melkite (Byzantines from the Middle East), Ukrainian, and Romanian. There are a few Russian Catholic and Italo-Albanian parishes or missions.