The Priest is the main alter attendant at the Church / Endmost Bethel. The Priest's job is to offer spiritual guidance to travelers, as well as assist them with associated tasks such as Praying, self-improvement, and bestowing blessings with Auras, the latter of which is done in exchange for a donation.
It's not specified which set of beliefs or religion the Priest promotes, but he often speaks of a "greater power", meditation, reflection, and prayer. Although the Church can be visited from anywhere with a Blue Save Flower, it's likely the Priest lives on-site, alongside the Keeper of the Star Map in some sort of rectory, as the island apparently contains more than just the grand cathedral hall visited by the player.
The Priest helps the player raise Class and Level in Baten Kaitos: Eternal Wings and the Lost Ocean, while in Baten Kaitos Origins he helps raise Class and exchanges Auras for money. While the Priest appears in both games, it's not clear whether or not it's the same person.