Rites within the Catholic Church:
A rite in the Catholic Church is the liturgical, theological, spiritual, and disciplinary heritage—including specific manners of worship, language, and customs—by which a particular Church sui iuris (autonomous church) lives out its faith in communion with the Pope. While the Latin (Roman) Rite is most common, there are 24 distinct, equal-status rites in the Catholic Church, comprising one Western and several Eastern traditions.
Below we list just a few of them.
Eastern Rite Catholic churches in communion with Rome:
They recognize the supreme authority of the Pope while retaining their own hierarchies, led by Patriarchs, Major Archbishops, or Bishops. They follow Eastern traditions such as Byzantine (e.g., Ukrainian, Ruthenian), Alexandrian (e.g., Coptic, Ethiopian), Antiochian (e.g., Maronite, Melkite), Armenian, and East Syrian (e.g., Chaldean, Syro-Malabar).
They share full sacramental exchange with the Latin Church, meaning Catholics of the Roman Rite may receive Communion and fulfill their Sunday obligation at Eastern Catholic parishes.
Syriac Catholic Church: a distinct Eastern Catholic church in full communion with the Pope, having separated from the Syriac Orthodox Church in the 17th-18th centuries.
Eastern Rite Catholic churches not in communion with Rome:
A communion of self-governing (autocephalous) churches that share sacraments with each other but not with the Pope. They accept the first seven ecumenical councils but reject papal supremacy and later Roman dogmas.
Examples include the Ecumenical Patriarchate of Constantinople, Russian Orthodox Church, Greek Orthodox Church, Syriac Orthodox Church (an Oriental Orthodox church that separated in 451 AD).