After the fall:
If you're interested I highly recommend the book Medieval Christianity by Kevin Madigan, it's a good read and not too inaccessible. (I've been on askhistorians a lot using this book!)
For hundreds of years the popes in Rome occupied themselves with maintaining their power in the Italian region and bringing secular leaders into the fold of "orthodox/catholic" christianity rather than some variant such as Arianism. During this time, they also often entreated the eastern roman emperor to intercede on their behalf and save rome from some threat or another (Lombards, etc.).
In the medieval times, the popes were constantly struggling against, and aligning themselves with, secular authority. More than once, the pope invited outside powers (the Ostrogoths, or the Franks) to keep the peace in the Italian region in exchange for blessings or favors.
It wasn't until the 12th century that popes were considered, by and large, to have authority exceeding that of regional authorities. Before the "Gregorian reforms," most of the church officials that most people would interact with (remember most people are rural farmers) were appointed by lay lords or were otherwise locally decided. The pope never really had total control over the area of "Germany," for example, but over time the pope and the Catholic hierarchy in general became more and more powerful in Italy, France, Iberia, England, etc.
One of the major turning points was the dictates of the pope, (11th C.) which asserted papal authority over kings and other Christian authorities (such as the other patriarchs). While these powers were not necessarily respected by contemporary secular authorities, it does represent a change in the attitude of the Vatican - that they had worldly as well as spiritual authority.
It was really the period of reform starting in the 12th century that put the papacy down the road to the ultimate spiritual and worldly authority. However, many of the powers ascribed to the pope were not explicitly enumerated until the "high middle ages."