I would say that the Catholic Church had political power over people in the Middle Ages in a few different ways.
First of all, there were places that were directly under the control of the Church -- places where the Pope was the official ruler of the country. This was in the Papal States in what is now Italy.
Second, the Church owned a lot of land in other parts of Europe. Each monastery, for example, owned lots of land and the abbots of the monasteries ran those lands. The people on the land (if they were serfs) belonged to the monastery. In that way, the Church had political power because it had temporal power over its estates.
Finally, the Church had moral authority that could be used politically. The Church was seen as the only way to Heaven and so the clergy could be pretty important in making people behave in certain ways. They could threaten, for example, to excommunicate people who did not make the "right" choices.
This gave the Church more power of an indirect sort.