All spells in Skyrim, except for racial powers and shouts, are part of magical schools. These are as follows:
Alteration: This rather broad school of magic consists of skills that, well, alter stuff. There are a number of shield spell effects available in this line, as well as spells that let you walk on water or pick up items from a distance. Alteration is rarely considered a caster's primary school of spellcasting, because there's nothing offensive about it, but many casters rely on shield spells from the alteration school to protect them from the blades of bandits and the fiery breath of dragons.
Those interested in using Alteration spells should note that there is a three-tier perk called Mage Armor that increases the effectiveness of defensive spells in the Alteration school if the player character is not wearing armor.
Conjuration: Governs the use of spells that conjure creatures or items. A conjurer can raise zombies to fight for them, or can summon bound weapons for defense. If you like the idea of a "pet class" the conjurer is just about your only route to take in Skyrim, as they're the only characters that can constantly bind creatures to their bidding.
In my opinion, this is a good school of magic for an offensive spellcaster, as the bound creatures can take pressure off of your character while also dealing damage to opponents.
Destruction: This is your straight-up, fireball-castin', lightening-rod shootin' school of magic. It's the one school of magic that virtually every mage character in Skyrim takes. If you want to challenge yourself, try building a mage who never, ever uses Destruction magic.
Destruction is largely based on elemental damage, and there are perks in the skill tree that offer enhancements to certain elements. This has pros and cons. Against some enemies you will be very strong, because they may be weak to an element you prefer. But on the other hand, you may find an opponent who is resistant, and therefore much tougher. Don't forget to switch up your offensive spells if you encounter an opponent who seems to be resistant to a particular element.
Illusion: Consider this the crowd-control school of Skyrim. This is where you will find spells that fear, charm, or command targets, forcing them to disengage or turn to your side. These spells are extremely useful in many dungeons, where you often fight more than one enemy and can turn them against each other. But they're not that useful in boss fights, where they can only buy you some time.
Other utility spells flesh out the line and make this a favorite of thieves. Invisibility is part of this school, as is silent movement. These make Illusion the favorite of mage-thieves. At skill level 50, it's possible to grab a perk that makes all of your spell silent, which is again of great use to mage-thieves.
Restoration: This is the school that controls healing spells and can also be used to cure certain conditions, like diseases. It can also be used to Turn Undead.
The usefulness of this school will depend on your playstyle. Because this is a single-player game where you can control one character, there's never a huge need to keep others healthy. However, this school can be useful for keeping you alive if you choose to play a sort of battlemage - a mage who gets up close and personal with AOE spells, for example. Although some mages choose robes, others choose heavy armor, fireballs and health spells to get them through combat.