A priest is any commander with levels in the holy magic path. They are always sacred and can be blessed, can preach to increase dominion, and priests with 3 or more can claim thrones. They are critical in battle to bless sacred units.
Many nations have priests with levels in other magic paths as well; EA T'ien Ch'i and MA Agartha, for instance, have only mage-priests. Since nations recruit many mages, this means that these nations will have no shortage of priests on hand. Others maintain a separation; EA Atlantis and Bandar Log have no priest-mages, meaning they must recruit dedicated priests if they want them.