Though it's worth pointing out that the Latin church and the Orthodox church evolved from the same system.
The Pope is technically an honored Patriarch.
There are more esoteric titles than the ones listed above, but the one's listed above should serve.
One important way to envision the medieval church is as the successor to the Empire. As the Wester Empire fell further into the past the Church it had established remained and took over many of the Empire's beauracratic functions.
Further, since Bishops were critical to the Catholic faith and were also fundamentally associated with cities in many areas they worked to stunt the rate of de-urbanization or create new cities.
Bishops would be in charge of Cathedral, technically a cathedral is the Bishop's seat, but the situation can be a bit more complicated in that you often have priests and/or deacons who assist the Bishop and often become the actual goto guys in charge of the Cathedral and its associated charities, lands, schools, courts, and institutions.
The normal title for the guy who was the bishop's assistant was Arch-priest. Though an Arch-priest can also be someone who runs a Mother church. That is a church that can fulfill many of the same functions as a Cathedral for a rural district that is too difficult for a Bishop to go to with any regularity.
Any given rank of clergy could be in charge of a shrine or none could, it is not uncommon for a great deal of faith practice to be run by the laity. Though there is certainly greater prestige in being run by a clergyman.
The Catholic Encyclopedia linked to below is a fantastic resource for any church related issues or even for medieval history generally.
http://www.newadvent.org/cathen/10285c.htm
That should give you an initial guide on how to use it.
Most of the drama surrounding the church of the period, particularly from an adventurers standpoint, revolves around the follow themes:
Security Issues: as they were frequently poorly defended or a lonely power, church and monastic properties were also frequently a target for raiders, profiteers, or local anger. Also belonging to this theme would be issues of more global security. The church of the period worked very hard to bring order to chaos, so they frequently brokered peace treaties and powerful institutions would bring unprotected or disorganized properties under their control. That's how you end up with Bishops running their local cities and Monastaries running small empires. They also tended to work on quality of living issues so social disruption and, for lack of a better term, land and environmental management.
Internal Issues: there was a lot of dispute inside the church over individual priveleges, responsibilities, aesthetics, theology, and property. An extensive but complex and variable legal system existed within the church to regulate this, but ineffeciencies were frequent and resulted in a great deal of debate. Further, the process was deliberately slow, the results of poor decision would be catastrophic and a delayed could frequently take care of the problem one way or another. Most stories of church corruption fit beneath this issue.
Popular Faith: this covers everything from the many popular religious movements of the day to miracles, extensive festivals, and crusades. A lot of adventuring goes here, finding relics, abolishing demons, confronting heretics, dispersing or blessing mobs, theatre performances, and just about everything under the sun.
High Faith and Reason: this is what the church of the period was most famous for, learning. Here are the great philoshopers, the schools, the work of the monastaries, the art, the literature, the mysticism, and the theology. This also covers the major major church reforms, the great councils, the fights with the emperors, schisms, the establishment or suppression of new orders, the begining of the crusades, the massive legal issues, and the establishment of things like rules of celibacy. Many issues of this type will in fact be extensions of the other areas save they are applied to the church at a much higher level. Your local church would respond to the other issues because they were immediate problems, this is the type of issue your local church responds to because it's been forced upon them or they are forcing it on the church as a whole.