The only classification I know of is black hat, grey hat, and white hat. Some people say the grey hat isn't even real, but I suppose it is all based on how you look at it.
Black hat: Bad boy hacker who does bad things such as stealing credit card numbers
grey hat: def 1- ex black hat, def 2- prankster hacker who hacks people for his own amusement
white hat: hackers that test out security software. They're paid by companies like Norton and McAfee to try to breach their software and report any holes.