Between 600 and 1650, the popes were the leading authority figures of Western Europe. In what ways did they assert political power?
During most of this time period, the Catholic Church was the one spiritual authority in Western Europe. Popes were the unquestioned (except for times when there was more than one pope) heads of this Church. They tried to use their spiritual power as a way to get political power.
This can be seen most clearly, perhaps, in the Investiture Controversy of the late 11th century. In this controversy, Pope Gregory VII was trying to take the power to choose church officials (such as bishops) away from the secular rulers. In doing so, he argued that the pope could depose any secular ruler that tried to keep the power to appoint church officials. This was a clear example of the pope trying to use his spiritual power to gain political powers.
We can also see this in the controversy between Pope Boniface VIII and King Phillip IV of France. The king wanted to be able to tax Church establishments in France and wanted clergy to be subject to secular courts. Boniface resisted this and excommunicated the king as a way of trying to assert his power. Here again, we see a pope trying to use spiritual authority to gain political power.