Catholic Church
Throughout the Middle Ages and prior to the Reformation, the Catholic Church was the main form of religion practiced by people in Europe.
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 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 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 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.