Priest Magic is better known as Divine Magic. It manifests in the form of spells and spell-like effects. Divine spells are drawn from divine forces (deities and demi-gods) and are considered generally less dramatic and destructive than Arcane Spells (Wizard Magic), with a greater focus on healing and protection.
Priests cast spells by petitioning their god or drawing power from the environment around them.
Unlike Wizards, Priests do not possess or require the use of a spellbook. Their power comes directly from the gods, through a process rather similar to praying or concentrating on the divine power that surrounds everything and everyone.
A Priest in the process of casting a spell can generally be identified as someone going into a ‘trance-like’ state. This might be for a split second or for several moments. It really depends on the complexity of the spell combined with the level of the priest.
Priests start out by knowing level 0 spells. As they level up through the game they are given assess to a larger list of spells ever growing in power. On the whole, most Divine Spells are not nearly as destructive as Arcane Spells, though can be extremely powerful and at times can be deadly. A powerful Priest is someone you do not want to get on the wrong side of.
Remember, although the casting act is generally automatic, there is always a chance of a critical or fumble and if the target unwilling the effects of the spell could be avoided or reduced via a saving throw.
Take a look at the full list of known Arcane Spells in the Grimoire Manual.