"Intristic" magic
Intristic magic is the Discworld's "standing magical field" and is basically what allows a flat planet on the back of a turtle to even exist. The other varieties of magic are methods of shaping this force. It warps reality in much the same way as gravity warps space-time. Areas with larger than normal quantities of background magic tend to display unusual qualities, even for the Disc.
There are eight colors in the Discworld light spectrum. The eighth color is "octarine", the color of magic whose existence is due to the magical field around the Discworld. In fact, eight is a number of some considerable occult significance on the disc and must never, ever, be spoken by a wizard. Instead they prefer to refer to it as "the number between seven and nine", "the number which is twice four", etc. Octarine is invisible to most of the people but wizards, and shines wherever some magic energy is consumed.
Very high quantities of magic (of the sort that might be produced by a "sourcerer") can knock a hole in reality, leading to an invasion by Lovecraftian monstrosities from the Dungeon Dimensions.
Wizard magic
Wizard magic is known to be taught at the Unseen University in Ankh-Morpork, Bugrup University in XXXX and Krull University in the secretive nation of Krull. It is very much a scholarly study, largely (many believe) to prevent anyone outside the universities realizing how easy it really is.
In fact, the older wizards tend not to understand how magic actually works at all, instead relying on centuries of knowledge to achieve their effects. Younger wizards enthusiastically experiment, pushing back the boundaries of knowledge and making new discoveries about the nature of the universe. They don't understand how magic works either, but have much more exciting words to explain why not.
Really, wizard magic is telling the universe what you want it to be like, in terms it can't ignore. This is very draining, due to the Law of Conservation of Reality (which states it takes the same energy to do something with magic as it would to do it without magic). This is why most wizards store magic in a staff. In fact, the most complicated part of most spells is to ensure the wizard survives.
Wizards can also keep spells in grimoires. But the problem is that a spell is still a spell even when imprisoned temporarily in parchment and ink. It has potency. This is not a problem while the book's owner still lives, but on his death the spell book becomes a source of uncontrolled power that cannot easily be defused. A temporary solution is the construction of large rooms made of denatured octiron, which is impervious to most forms of magic. Here the more critical grimoires can be stored until their potency has attenuated. (This is how the Octavo, greatest of all grimoires, formerly owned by the Creator of the Universe, came to be at Unseen University).
A "sourcerer", on the other hand, creates his own magic, and can therefore do just about anything with no effort. As the above notes on intristic magic suggest, this is very dangerous, which is why there are no sourcerers any more.
Witch magic
Witch magic is taught on a one-to-one basis by older witches to apprentices. Although magical talent tends to run in families, witches do not teach their daughters, feeling that this would cause a sort of magical inbreeding.
Witch magic is subtler than wizard magic, generally encouraging things that happen naturally or working on people's minds so that they impose the effects themselves (in fact, a good witch can do this without magic at all. This is called "headology"). As a result it is less energy intensive, which means that a witch can do more than a technically equally powerful wizard. However the same Zen-like knowledge that gives them this ability generally discourages them from making a big deal about it, beyond refusing to take wizards seriously.
Discworld Voodoo is considered to be an aspect of witch magic, combined with a sort of do-it-yourself religion, relying heavily on the power of belief described below. The most powerful Discworld voodoo-women can deliberately create moderately powerful gods for a specific purpose.
Generally speaking, witches are women and only women and wizards are men and only men. The only known exception is Escarina.
The Power of Belief
This is very common on the Discworld, and is a useful "energy saver" in witch magic (and, to a lesser extent wizard magic). Essentially, if something is believed strongly enough, it is true. As mentioned above, witches often use this. For example, if you wish to turn a cat into a human, the easiest way is to convince him, on a deep level, that he is a human.
More significantly, it is also belief that gives the gods their powers. Discworld gods start off as tiny spirits, and gain power as they gain believers. This does not necessarily mean worshipers; a thousand people cursing you as an evil djinn has the same effect as a thousand people singing psalms in your honor: it generally gives you enough power to make some lightning, which is a very common (and proved to be effective) way to show that you are a god.
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... весели, засмени, румени, червени!
