Destruction is used to create elemental attacks, thus being used as an offensive form of magic to defeat foes. It is almost essential to any mage that plays an offensive role. Generally, destruction magic is split into three types of spells: fire, frost, and shock.
Restoration magic is focused on creating and harnessing life force, thus it is commonly used for healing wounds. It also has spells called Wards, which can be used to reduce the damage inflicted by enemy spells, just like a shield would block an enemy blow. Restoration also has a series of spells used to turn and/or even harm the undead.
Alteration spells change reality in a specific way. Alteration spell can create magical barriers to act as armor, as well as cast Waterbreathing, Telekinesis and Transmuting. Alteration isn't a specific set of spells, but rather a wide variety of spells each with unique effects.