In Shogun: Total War, a priest is an agent avaliable to be recruited by a clan by converting to Christianity upon contact with foreign traders. They are the only type of religious agent in the original Shogun: Total War, and the first example of a religious agent in the Total War franchise.
Priests are the agents created to deal with civil unrest caused by Buddhist and Shinto populations should the player chose to opt for Christianity. They replace emissaries and have the same effects as them except for being able to reduce civil unrest in your own regions.
Recruited from Churches.
Priests came to Japan from the Europe in the middle ages as missionaries of faith. In the 16th century, Dutch and Portuguese traders brought Protestantism and Catholicism to Japan. Usually in return for their trade services European missionaries were given free entry to Japan much to the dislike of many of the native faiths, clashes were quite common.
These Jesuit-trained Priests are excellent Emissaries when dealing with Christian Daimyo. While sending a Priest is no guarantee that a proposed treaty will be accepted, at least you won't get his head sent back in a bag. Unless the enemy Daimyo is Buddhist.