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 very difficult to re-enact within the SCA.
The Catholic Church is a heirachy, with the Pope at the top, then Bishops or Archbishops, and so on down to your local parish priest. It also has Monastries, which sit to one side of the structure, the most famous of which was Cluny.
The Church is the 'those who pray' division in the Medieval order of those who work, those who fight and those who pray.
The Church is notable in that the Pope is elected by the College of Cardinals (usually. Having an army in Rome helps your candidate get up though).
The Pope often gets into arguments with ther 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.