Priests, bishops, alims, cardinals, and imams all work to convert province populations to their respective religion, so they will have an effect on the religious percentages. The religious makeup of a province will affect the happiness. These agents can also be used to request alliances or ceasefires.

Zeal is increased by the presence of inquisitors, grand inquisitors, and imams. Zeal is important when preparing the way for crusades and jihads. The higher the zeal, the more recruits the marker will pickup while it sits in the province. Higher zeal also gives the inquisitors a better chance to try someone for heresy. If you play an Orthodox faction, zeal is fairly worthless and you probably want it as low as possible everywhere.

Religious percentages, zeal, and governor's piety are tied together to affect province happiness, but I don't know the exact formula. Of course, there is a problem if the faction religion isn't dominant within a province. There is also a penalty if the zeal of a province is high and the governor's piety is low, or vice-versa.

I generally don't play the game with a heavy focus on religion, but you do need at least the basic priests/bishops/alims running around converting newly conquered territories. If you want to use crusade/jihads heavily or hammer your Catholic rivals with the Inquisition, zeal becomes a lot more important.