Anyway, here are a couple of mine.
Altmer Apprentice Illusionist (that is, using Illusion, and one or two other magic schools). This type of character poses quite the challenge when fighting other mages as it would be quite easy to be killed with a well placed fireball. To get around this, you are forced to rely very heavily on Illusion (my favourite magic school) which requires some thought.
Healer. Some of you may know that my favourite character is Helena Aurelie, a Breton Healer born under The Apprentice. Her main magic schools are Restoration, Illusion and Conjuration (only Daedra) though she uses a few spells from the other schools. She won't deal any direct damage to any living thing (except Necromancers). I like this as it leaves a few options that might otherwise not have been thought of. For example, Helena sometimes infiltrates forts under Invisibilty, sneaking past everyone. Other times she commands her enemies to duel between themselves, so she doesn't have to do anything.
So...share yours!