So basically Vanus Galerion invented 'schools of magic' by looking at all spell effects known to him and grouping together the ones he thought here cast in a similar manner. The original grouping ignored spell effects in the groups, and it just happened that each school contained effects of similar type (although it makes sense, a way of thought meant to destroy is likely the base of any destruction spell, with the chosen element simply caked on top.) This is why each school is a skill; if you are proficient at casting one illusion spell you basically have it down for most of the others.
So here is my breakdown of the schools:
Alteration Making a fantasy for yourself
We are lucky, Reality and Other Falsehoods directly tells us how Alteration is cast.
"To master Alteration, first accept that reality is a falsehood. There is no such thing. Our reality is a perception of greater forces impressed upon us for their amusemant... To cast Alteration spells is to convince a greater power that it will be easier to change reality as requested than to leave it alone. Do not assume that these forces are sentient. Our best guess is that they are like wind and water. Persistent but not thoughtful. Just like directing the wind or water, diversions are easier than outright resistance. Express the spell as a subtle change and it is more likely to be successful. "
So basically you are persuading physics to break itself without literally asking water to let you walk on it. So how do you do it? One convinces HIMSELF that you have persuaded water to let you walk on it. Or rather, that you naturally would be able to, it's water isn't it? from Breathing water, "Seryne pointed to the small silver fish darting along the water's edge: 'They don't find it so. They breath water just fine.'
'But that's not magic.' 'What I'm saying to you, boy, is that it is.'"
So alteration is largely making a fantasy for yourself and then adding magic to it; the raw potential energy will make your reality exist – at least for a while.
Illusion: Magic to alter an opponent's perspective of reality
In terms of what they do, the main difference between Illusion and Alteration is that Alteration actually changes physics while Illusion just makes the target percieve things differently.
“You see, it's all concerned with magicka's ability to alter the perception of objects without changing their physical compositions. Removing sensual data, for example, to cast darkness or remove sound or smell from the air."
“Certainly,” smiled Massitha. “Nothing changed in the vampire's form, except its ability to move. Like I said, it's a very useful School.”
I believe Illusion to be tampering with your target's perception of reality. It is in fact the opposite of Alteration; in Alteration you alter your own perception of reality and then apply magic. With Illusion one uses magic to alter an opponent's perspective of reality. Think of the Eragon books, and the magical mind-reading in that is basically how I view illusion in TES.
This brings up a good point though; with enough willpower and mental fortitude someone could ignore an illusion spell such as Paralyze.
Thaumaturgy: unknown
Very little is known about Thaumaturgy and it only appears in Daggerfall. I would imagine it is somewhere between Illusion and Alteration; it's spells were split between those schools.
Destruction: Raw magic
Destruction is probably also somewhat similar to Illusion, Bero would have needed some reason for suggesting they be merged. However, on it's own, I can find no quotes about how it works. Left to guess, I have come up with my own theory. Where as in Illusion the Magica is applied to alter someone's perception, in Destruction a spell might start out similar and then have the magica applied directly. We know that light is a form of magica in TES, so lightning and fire are both very crude versions of it. Further, a frost spell could simply be doing the opposite of a fire spell.
When it comes to damaging armor or stats, the addition of potential energy with very few instructions – not even light on fire or shoot like a bolt – would surely matter. The skill in these spells is applying magica to the right thing.
Mysticism The old way; wild magic
One of the oldest forms of refined magic in TES.
"Mysticism seems to derive its power from its cunundrums and paradoxes; the act of experimentation, no matter how objectively implemented, can influence the magicka by its very existance. Thus, the Mystic mage must regulate himself to finding consistant patterns in an imbroglio of energy. In the time it takes him to find a source with a consistant trigger and result, his peers researching in other schools may have researched and documented dozens of new spells and effects. The Mystic mage is a patient and uncompetitve scholar."
“The common man looks at an object and fits it into a place in his way of thinking. Those skilled in the Old Ways, in the way of the Psijic, in Mysticism, can see an object and identify it by its proper role. But one more layer is needed to be peeled back to achieve understanding. You must identify the object by its role and its truth and interpret that meaning."
My theory on this has largely been impacted by that it is the oldest known form of magic. A mysticist feels the natural flow of magic in the world
Enchant
"'A simple spell cast once, no matter how skillfully and no matter how spectacularly, is ephemeral, of the present, what it is and no more,' sighed Magister Ilther. 'But placed in a home, it develops into an almost living energy, maturing and ripening so only its surface is touched when an unskilled hand wields it. You must consider yourself a miner, digging deeper to pull forth the very heart of gold.'"
Adding magic to an object, implanting it within.
Conjuration Control.
From these I have gathered that, with the application of potential energy (magica) the easiest part of conjuration is the conjuring. An apprentice can summon an extremely powerful daedroth from oblivion, and a journeyman could probably summon things to him from Akavir. This is done by reaching out mentally and connecting with what you want to summon (you can only use conjuration on living things, inanimate objects fall under mysticism.) If it wants to come, or you can force it to do so with or without a protonymic, it will. This is why master conjurers form a mental link with eachother; they are experts at reaching out with their minds.
Sources: http://www.imperial-library.info/content/oblivion-doors-oblivion,
http://www.imperial-library.info/content/frostfall-3e-432
http://www.imperial-library.info/content/morrowind-frostfall-book-ten-2920-last-year-first-era
Restoration
Honestly i've pored over the In-game books but I CAN'T FIGURE THIS OUT
Necromancy Rituals.
The raising of the dead. Honestly this is the simplest school and all i had to go from was http://www.imperial-library.info/content/morrowind-preparation-corpse.
The caster puts magic into the target, raising it since the target really is just lacking a soul and souls are a form of extremly unrefined magic.