The Catholic Church is a hierarchy, with the Pope at the top, then a confusion of offices, including Cardinals, Bishops or Archbishops, and so on down to your local parish priest. It also has Monasteries, which sit to one side of the administrative structure. The most famous of the monasteries was Cluny, influential in implementing Church reform.
The Church constituted the oratores, 'those who pray' in the three groups of the Medieval order, divided into those who work, those who fight and those who pray.
The Pope often got into arguments with the King of France or the Emperor of Germany over their respective rights, for example whether the State has the right to tax the clergy, and if so does it need their consent.
Occasionally, there were two (or more) men who claim to be Pope. This Is Bad. Those who eventually lose were known as anti-popes.
Roleplaying the Catholic Church
The SCA chooses not to practice religion in an organised sense, and in particular the articles of Corpora prevent the forcing of any person's religious beliefs on any other person within the SCA. Therefore the institutions of the Catholic Church or any other Church are difficult to re-enact within the SCA.
That said many do this successfully in an unofficial capacity (like with households) forming monasteries, abbeys and the like. Monk's and Nun personas are growing more common while those playing "Bishops" are quite rare. A persona based upon the Clergy of the Church is both a viable and valid persona.