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Catholic can be an adjective or a noun.
The adjective Catholic means universal, therefore a Catholic Church considers itself the universal church.
Traditionally the Catholic Church is seen as the Roman Catholic Church and Roman Catholics often insist on calling themselves just Catholics because they think they are the only universal church of Christ and the connection with Rome is secondary. When Catholic refers to the Roman Catholic Church is is written with a capital. It also refers to the Eastern Catholic churches, which form part of the Roman Catholic Church, yet follow Eastern rather than Roman traditions in liturgy and other matters
The other strong claimant for the "Catholic" title is the Eastern Orthodox Church. The Orthodox Church and Roman Catholic Church, which comprised the seemingly "Universal Christian church" before then, officially fell apart in 1054 during the Great Schism, although the process had been going on for centuries, since the fall of the Western parts of the Roman Empire during the 5th century and the developments in the Eastern Roman Empire, also known as the Byzantine Empire. Theologically, Orthodox and Roman/Western believers usually accept that they are both fundamentally part of the One, Holy, Apostolic and Catholic church (The Four Marks of the Church as defined by the Nicene Creed) but the other party just needs to reject some little heresies and adjust their views slightly to be in full communion with their "brothers and sisters in faith" on the other side.
Naturally other Christians disagree that only Roman Catholics and/or Orthodox Christians are part of the universal church. Lutherans and the Church of England also see themselves as Catholic or part of a universal church as do other churches that follow the Nicene and Apostles' Creeds.
The term Catholic can also be used to refer to splinter groups which have broken away from the Roman Catholic Church. It is used in particular to refer to those groups which while breaking away maintain much of the Roman Catholic church's traditions - as opposed to Protestant churches, which rejected much of the Roman Catholic Church's traditions when they split from it.
High Church outside Roman Catholicism
Catholic also refers to a tendency in Anglicanism (Anglo-Catholicism), which sees the Anglican church as nearer to the Roman Catholic Church than to Protestantism, and thus adopts much of the Roman Catholic church's tradition (including belief in purgatory). A similar tendency also exists in the Lutheran churches (Evangelical Catholicism).
The noun Catholic means a person who believes in Roman Catholicism, who used to believe in Roman Catholicism or who was at some time baptised into the Roman Catholic Church, perhaps as a baby but never believed it. Catholics have a doctrine that once a Catholic always a Catholic.
The term Catholic can further refer to the followers of one of the other churches that call themselves Catholic, such as the Old Catholic Church, the Polish National Catholic Church, or the Palmarian Catholic Church.