The Wizard class has 4 specialist types (as well as a plain Wizard) which are Demonologists, Elementalists, Necromancers and Illusionists.
The types are fairly self-explanatory - they mean what you think they mean.
To go with the types of magician there are types of spell. There are Petty Magic Spells which anyone can cast, and there are the battle magic spells. All the battle magic spells have a rating from 1 to 4, corresponding with the minimum level required by a spellcaster to cast the spell. (Except Alchemists, who aren't proper magic users, so they can cast spells with levels of up to one less than their Magic Level). Here is a summary of what types of spellcaster can cast what types of spell:
The way the system works ties in with the rest of Warhammer is that each type of magician has it's own set or careers (ie Wizard's Apprentice, Wizard Level 1, Wizard Level 2,...). These are treated slightly differently from normal careers as to move from one level to the next you need to complete the current level. This means taking all the advances to your profile, all the availible skills, and at least two (if available) spells from that level. This also includes the Apprentice careers.
You do not have to choose which type of magic user to be and stick with it - generally you can change around - be a normal Wizards up to level 4, then complete up to level 2 of Illusionist then move on to become a Necromancer. The only restrictions are than to start on a specialist Wizard career, you need to have completed at least Wizard Level 1, and that Elementalist and either of Necromancer or Demonologist are mutually exclusive.
NB you only raise your Power Level when you increase the maximum career level that you have attained, ie having completed up to Wizard Level 2 and then starting on the Elementalist career path you will only increase your Power Level when you reach Elementalist Level 3.