Certain faiths (details on which, other than Catholicism, to follow in a later diary) replace the court chaplain with a bishop. If you still have a court chaplain, then they’ll behave much like other councillors, though some faiths may still have a harder time firing them.
Bishops use a mechanic called leasing (though they’re not the only ones to do so), whereby they control all levies and receive all taxes from every temple holding in your personal domain. Additionally, your bishop will receive a fraction of the taxes and levies generated by all of your vassals’ bishops. All told, that’s a lot of ducats the Church seems to be getting, isn’t it?
Of course, as loyal subjects of the crown, your bishop will be very happy to hand over taxes and troops to you, scaling with quite how happy they are. A loyal bishop is a huge boon for your economy and military, and can make the difference between unstoppable royal might and economic ruin.
A recalcitrant bishop, by contrast, is an utter pain. If they do not approve of you at least a little, they’ll hold back taxes and levies until you meet their standards again, and if they actively hate you, may even begin conspiring with others to replace you with a more pious monarch.
Of course, you’re probably asking what stops you from firing your bishop and replacing them with, say, a good friend of yours?
Around these parts, that’s what we call heresy. And you’ll have to wait till the religious dev diary for details on exactly what your options are for legally sacking your bishop. As for illegally, well, no Pope can stop a knife to the base of the spine...