Sages and scholars tend to classify magic into a bunch of different, occasionally-overlapping categories kind of like the biological classification of life in our world. Never forget that King Phillip came over for grape soda! He didn't find any grape soda, though, so he called a few demons to destroy everybody. This act would go down in history as the "little ice age". But I'm getting off-topic. Anyway, we've got some occasionally-overlapping magical kingdoms:
Conjuration
This one is fairly self-explanatory. You are usually making something from nothing. Or you're pulling something that already exists to you. These would be different Phillips, err, Phylums. So what can you do with conjuration? What sort of limits are there?
Conjuration tends to be both the easiest and toughest magic to pull off at the same time. Making a ball of fire explode in the middle of a crowd of people falls under King Conjuration, and that's mid-level magic. Making my hot drink icy-cold is the same sort of conjuration, and it's pretty easy. They are both the same idea, working with energy. It's easy to work with energy, especially if you're a sorcerer (sorcerers find it really easy). Some scholars theorize that this is because there are infinite planes filled with energy out there, and every time you use energy you're just pulling it from them. Of course, some scholars also theorize that the second moon is made of blue cheese, so that's about as much stock as you should put in their opinion.
Conjuration also includes such stuff as summoning and calling various creatures, including demons*. For whatever reason, powerful mages tend to have an easier time with this than sorcerers. Maybe that's because doing so involves very exact formulas of magic, kind of like how mages cast spells in general. At least, that's what the scholars say.
There is also the standard "making something from nothing" thing, like conjuring great towering walls of iron. Or stone. Or invisible-but-still-existing-and-impassable force. Or sheep. Or eyeballs. This is actually fairly difficult, and almost always temporary. A really powerful mage would be able to make their iron or stone stay there permanently, but never their sheep or their eyeballs (it has something to do with the organic vs. inorganic nature, apparently). See: calling demons.
Necromancy
Well, everyone tends to know the basics of this one. You uh, make zombies. And wraiths, and specters, and vampires, and nightshades, and, well, if you can think of any sort of undead, you can probably make it. Depending on how strong you are. While zombies are generally pretty mindless, the intelligent undead basically only exist to do things like kill people and feed on their corpses, so necromancy gets a pretty bad rap and nobody really practices it.
Necromancy is more than making undead, though. Anything that involves draining somebody's life-force is necromancy. Presumably, restoring somebody's life-force would also be necromancy. I say "presumably" because while there are healers, healing tends to work in such a way as that the body rapidly speeds up its recovery to close-to-instantaneous rates and, while that is necromancy, nobody's ever figured out a way to use magic to, say, make someone who is tired not-tired.
So basically, anything that directly involves life/death belongs to necromancy.
Transmutation
You change one thing into something else. Yaaaawn. Or, you bend existence or reality. Anything that involves teleportation would fall under this. The thing is, nobody can really teleport. Well, people are pretty sure that dragons can do it. And if dragons can do it, then sorcerers probably can too, because sorcerers seem to be pretty good at the things dragons are usually pretty goo at. Sages theorize that that means that they access magic in pretty much the same way.
Some scholars have argued for the inclusion of fortune-telling or divination as a magical kingdom, but most of them think that that's just Conjuration or Transmutation.
So what is most magic, then? Well, most magic is something making street-lamps that pour light out continuously in the city. That's conjuration, and fairly easy conjuration at that because it's working with energy. It actually involves transmutation too, though, because a magewright making a streetlamp is picking up a rock or a piece of glass or something, and modifying it to be something magical: he or she is modifying the structure of the item to continually pull energy from whatever magical source it comes from and emit it. This is why it's difficult to cut off established magical effects like streetlamps: whatever they are coming from has been modified to make it constantly work that way. So you can't just form a knife out of magic and slash the threads (standard dueling practice to keep you from being hit with something that will kill you), they'll just be renewed after a little while when the rock or the piece of glass emitting the light pulls energy out of the stream and emits it again. Of course, you could just hit the rock with a sledgehammer. It's just a rock, after all.
* Being able to call demons is generally the cut-off point between weak mages and strong mages. If you're able to call demons, you're a pretty hardcore guy or girl.
Next: A sorcerer's view of magic.