A long post about the schools, and their ways of getting the desired effects out of spells.Conjuration is about intense focus and desire, according to this passage of http://www.uesp.net/wiki/Lore:2920,_Frostfall_%28v10%29
“My child,” she crowed. “You have come so far in so short a time. What were you thinking when you summoned this daedroth?”
It took Turala a moment to recall whether she was thinking anything at all. She was merely overwhelmed that she had reached out across the fabric of reality into the realm of Oblivion, and plucked forth this loathsome creature, conjuring it into the world by the power of her mind.
“I was thinking of the color red,” Turala said, concentrating. “The simplicity and clarity of it. And then -- I desired, and spoke the charm. And this is what I conjured up.”
“Desire is a powerful force for a young witch,” said Mynistera. “And it is well matched in this instance. For this daedroth is nothing if not a simple force of the spirits. Can you release your desire as easily?”
Turala closed her eyes and spoke the dismissal invocation. The monster faded away like a painting in sunlight, still blinking confusedly. Mynistera embraced her Dark Elf pupil, laughing with delight.
http://www.uesp.net/wiki/Lore:Mysticism_-_Unfathomable_Voyage
Alteration's already been covered.
Illusion is all about how you change something's perception without altering it physically. According to "http://www.uesp.net/wiki/Morrowind:Incident_in_Necrom"
“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. It can help by--”
Restoration (at least fortifying) is more about using Magicka to help yourself or others in general, using various ways and methods, according to "http://www.uesp.net/wiki/Morrowind:The_Four_Suitors_of_Benitah"
It took him some time to master it, visualizing magicka streaming through his body, pumping through the very fibers of his muscles for a time, giving him strength far beyond the puny power nature had intended.
He taught him how to entrench his mind for the sudden assault of awareness and aptitude that would assail it, how to give himself to the sudden thoughts and theorems that would invade his consciousness.
Oin agreed, and for the next several weeks, he learned the spell to make his endurance like that of the oldest stone. He learned how to shrug off the effects of frost, poison, fire, and charges of lightning, pulling the pain into a reservoir of magicka and expelling it.
Oin agreed, and Yakin taught him the spell that would fire his impulses with magicka. Over several weeks, he learned how to supplant his own natural energy with the spell's, how to view the world at the slower pace a man with advanced agility sees.
As for destruction, I've got no clue. However the people in the Destruction Skill Books seem to be very merciless and angry, it's also mentioned in http://www.uesp.net/wiki/Lore:The_Refugees, how a healer reversed her spell when she became angry.
"Near on twenty years, before this whole black, bloody business. I was Camoran Kaltos's chief scout, and Haymon was his sorcerer and advisor. I helped them both, when they were vying for the Camoran throne, and began the conquest of - Ouch!"
Rosayna has ceased her healing. With eyes of fury, she had reversed her spell, and the closed, mended wounds were opening again, dark infections returning. She held him with surprising strength when Orben tried to pull back.
"You bastard," the healer courtesan hissed. "I have a cousin in Falinesti, a priestess."