The wizards, are largely these days, the province of graduates of Unseen University, Ankh-Morpork. There are eight orders of wizardry and eight grades associated with UU, and in practical terms the affairs of academic wizardry as a whole are run by the ARCHCHANCELLOR and faculty.
Wizard magic generally consists of illusion, a little weather-making, fireballs and the occasional darning of the Fabric of Reality. Fundamental to its use is the wizards staff, usually about six footlong with the proverbial knob on the end. Daily rituals with the staff, accumulate magical power which can be discharged very quickly at need, or stored in spell books and triggered by the syllables of the spell.
The more senior the wizard, the less likely he is to do any showy or practical magic. Senior wizards time in the UU is taken up with sleeping, eating at least four large meals a day, University administration and generally, well, just existing as a wizard as hard as they can.