Did they have some semblance of political power? If not, what were their roles and when did the position become one of such power that a Pope could crown Charlemagne Holy Roman Emperor?
The Roman Empire slowly converted to Christianity. Only in 380 did Christianity become the official state religion. Bishops and patriachs would have been around before and many would've popped up after the edict. In the east, the patriarchs all "ruled" their respective areas, with the patriarch in Constantinople as their chief (please correct me if I'm wrong). In the west, the bishop of Rome (aka the pope) hadd always been the most important bishop (in the west), because Rome was Rome, ofcourse.
The argument about what kind of hierarchy the church should use (pope as supreme leader or a number of patriarchs) would eventually (again, over hundreds of years) lead to the Great Schism. Now there was not 1 Christian church, but an eastern Orthodox one and a western Roman Catholic one.
On christmasday in the year 800, the Pope crowned Charlemagne Emperor in an attempt to revive the Western Roman Empire and gain more legitemacy (sp?) for himself (or atleast his title) to become the supreme head of a unified Christian Church.
In short, the christians of western Europe accepted the bishop of Rome (or pope) as the head of their religion/church. When pope Leo III crowned Karl the Great Emperor, christians accepted it because of the popes status and title. At the time, the kingdom of Francia (Charlemagnes kingdom) was already a significant powerhouse compared to the states in area.
I also believe the pope requested Charlemagne to invade Italy, because the Lombard king (who ruled nearly all of Italy) was hostile towards the pope. The crowning could be seen as a reward of ridding the pope of the Lombard king.
I hope this wall of text makes any sense. Please feel free to ask questions or disprove me where I'm wrong.