Magical damage can be reduced through resistance.
Elemental damage refers to magical damage caused by fire, lightning bolts, and so forth. Many of the Bright Wizard's spells, for example, are a mix of elemental and other forms of damage.
An item that increases elemental damage: Champion's Sapphire Band (Touch of Frost)
Spirit damage represents magical damage that affects or comes from the non-material world, attacking your soul as much as your body. Certain careers, such as the Warrior Priest, have a lot of spiritually-based attacks, whereas others may have only a few or none at all.
An item that increases spiritual damage: Guardian's Iolite Band (Touch of Spirit)
Corporeal damage tends to represent natural magic, impact hits (physical), and so forth.
An item that increases corporeal damage: Sentinel's Amber Band (Touch of Earth)