What were the connections between the Holy Roman Empire and the church?

The Holy Roman Empire and the church were, as you might guess based on the name, quite connected entities in the beginning. The church was responsible for the crowning of Charlemagne, the first emperor who formed the actual Holy Roman Empire. However, in the centuries that followed, one of the emperors of the Holy Roman Empire got into an argument with the pope of the time because of which important consequences ensued.