The Papacy in the Late Middle Ages
Papal Government
Justice and Canon Law
It's misleading to think of regular courts of law in the curia. Rather, you should picture ad hoc committees formed by cardinals and other Church officials to hear specific cases. They were served by canon lawyers who did the research and provided technical counsel. Various offices were empowered to hear legal cases of different kinds.
The Datary heard petitions that did not require a decision in law. It also held dispensations from canon law (like marriage problems). The Consistory was the highest court of appeals. Special courts could be assembled for specific cases. The Rota (audientia sacri palatii) heard cases concerning benefices. This was a huge amount of traffic, for it involved all the arguments over who got what.
The Audientia determined technical legal points and investigated documentary evidence. This court also heard appeals when one party in a case objected to the choice of the judge. Appeals from this court went straight to the Chamberlain.
Any case could, in theory, be appealed directly to the pope. If he chose to hear a case, he usually constituted a special jury, usually comprised of some of the cardinals. The pope himself heard and decided only very exceptional cases (e.g., the question of putting Boniface VIII on trial posthumously).
The Penitentiary courts administered canon law with respect to ecclesiastical penalties. These courts heard requests for dispensations from penance, the commutation of vows, requests for absolution, and dispensation from censure. These were a tangle of courts, to which at least a semblance of order was brought by John XXII in 1338.
All court cases could impose fines, and this was an increasingly important source of income for the papacy. It was also one of the chief irritants to the rest of Christendom. Add to this the need to present "gifts" to those involved with your case, and the fact that cases could last for years, and it can readily be seen that at least some of the criticism levelled against the papacy had a large element of self-interest in it.