Magic originates with the Chang-Shouming, an ancient, mostly extinct race. They resemble the Chinese and speak the same language.
Wizards are descendants of this great race. The blood is diluted heavily because it is mixed with human blood.
The Chang-Shouming had an extra organ. This is what enabled them to use magic. Certain human descendants of the Shouming have it, too, thus there are wizards. This doesn't mean that their siblings or parents have the organ, too. (Kyle and Kenneth aren't wizards, for example).
However, human wizards need an implement specially enchanted to channel their magic. If magic is used without one, it is less potent and more likely to strain the organ. There are only two exceptions (the king and his brother; however, that is because they have such a high concentration of Chang-Shouming blood; their father was a full-blooded Shouming that came from a family known for its magical ability)
This organ can get tired (similar to muscle). Also similar to a muscle, a wizard can exercise it and make it stronger, increasing the potency of spells and their magical endurance. This organ, similar to the brain, doesn't finish developing till the early twenties.
A wizard's ability is limited by the stage of development his organ is at (I need a name for this organ...). There is only one possible exception, although it hasn't been confirmed by the scholars of magic: Godlings.
All the forms of magic are distantly related, of course, but only because of their origins. For instance, mind-bending is
completely different from necromancy, illusion, or, say, elemental magic. True, it is possible for an experienced master to replicate the effects of other magics with his area (for instance, an Illusionist creating the appearance of lightning and giving it an electrifying effect). However, for most our characters (the exceptions being Filirian and Vorxpol), it is nearly impossible to do so.
Other magics will be much more difficult than the students' specialty, which is already fairly difficult for them.
On the topic of Weather Manipulation and Mind-Bending:
WM is merely an extreme form of elemental magic and Mind Bending is an extreme form of telepathy, but involves much more than simple mind-reading. A master can completely take over the minds of others and even cause others to go insane. Both of these happened to Filrian; however, as he demonstrated, these effects aren't necessarily permanent. A master wizard can overcome these effects.
In order to be considered a master, one usually needs to be very experienced and a prodigy. The youngest to do this (besides a Godling) was 24 (Filrian); however, this was only because he was forced to by dire circumstances. He says that it would have taken another few decades had he not been pushed into it by captivity to a mind-bender. Normally, mastery is achieved later in life, usually 30s or 40s.
Just a note: All Court Wizards and Academy teachers are masters.
I hope this answers your question. If it doesn't, let me know