The Thaumaturgist can be thought of as a "battle priest". They are pretty good fighters, but specialize in the magical realm of Thaumaturgy. This calls upon blessings of the gods to "buff" party members. Whether that be to create a Spiritual Hammer, enlighten someone, or increase strength (and many more effects).
They also have some devistating spells like Slay Living (kills an enemy outright), Blade Barrier (position a set of whirling blades on the battlefield), and Righteous Spear (piercing damage to an enemy). They can even raise a dead party member! They are a great alternative to a Priest (prevention instead of curing).
Finally, Thaumaturgists have the skill to uncurse magical items, leaving behind any good special effects after removing the curse!
