Every ability, power or spell belongs to one of seven schools, which defines what can be done to absorb, immunize against or resist it. Your Resistance against five of these schools are displayed in your character information.
If you are a caster, and an opponent uses 
[Counterspell], 
[Spell Lock], 
[Kick], 
[Shield Bash] on you while you're casting a spell, these and other abilities will not only interrupt that spell but also to prevent you from casting any spell in that school for some amount of time. It's good to know what schools your abilities are in so you can switch to another when this happens. It's fairly clear for Mages (the school of the spell is the same as the tree it falls under) and Warlocks (if it doesn't involve fire, it's Shadow). For other classes it's not always clear. Fortuneately, the tooltip of any damage spell will specify the type of damage it does.
Physical
Not presented as magic to the user, but treated as a school by the game mechanics.
Attacking with a weapon falls into this school; the target can use Defense to avoid the attack, a shield to Block it, the Parry skill to block it with a weapon, and Armor to resist some of the damage if all else fails. Damage taken is Physical damage. Most Warrior and Rogue abilities are Physical.
Some physical abilities, like Demoralizing Shout and Thunder Clap, are treated like spells. These abilities have a basic resistance chance, but otherwise can't be avoided. This resistance can't be increased.
Arcane
Arcane magic typically inflicts direct damage.
Hunters have 
[Arcane Shot], and Druids' 
[Moonfire] and 
[Starfire] spells are their primary source of spell damage. Night Elf Priests can also do Arcane damage with [Starshards].
The true masters of the Arcane, however, are Mages. For Mages, the Arcane school is primarily a utility line rather than a damage line. Any spell from a Mage's Arcane ability category is of the Arcane school, including two spells that do damage ( [Arcane Missiles], 
[Arcane Explosion]) and a very broad range of utility spells: Polymorphs, 
[Arcane Intellect], 
[Mana Shield], 
[Mage Armor], 
[Blink], Teleports, Portals, [Counterspell], and Conjured Gems, Food, and Water.
Hostile Arcane spells (including those that do no damage) can be resisted; there are items that increase one's Arcane resistance.
Fire
Spells of the Fire school typically inflict damage, which can be direct damage and/or damage over time.
Many spells and environmental effects (such as lava) do Fire damage. The class most associated with the ability to control fire is the Mage. Fire is also one of the fundamental elements that Shamans can manipulate with their Totems. To a lesser extent, Warlocks can also deliver Fire damage. Warlocks specced heavily into their Destruction talent tree are sometimes called "Firelocks". Hunters also have traps that deal Fire damage.
A mage's 
[Fire Ward] and a shaman's 
[Frost Resistance Totem] are examples for non-damaging abilities in the Fire school. Like Arcane resistance, Fire resistance is a statistic that can be affected by spells and items.
Targets of Fire spells also have an increased chance to break out of Frost-based root effects.
Nature
The Nature magic School is usually used for nature based attacks, buffs, and curative spells. Nature damage can be direct damage (like lightning) or damage over time (like poison).
Acids, diseases and poisons inflict Nature damage, and damage from lightning is also considered part of the Nature school. Shamans' and Druids' healing spells are Nature spells, and other nondamaging spells such as 
[Entangling Roots] and 
[Faerie Fire] are in this school as well. Hunters' Stings are Nature based, as are Rogues' Poisons. There is also a Nature resistance statistic.
Frost
The Frost school is effective both for attack (doing damage) and defense (raising armor value). Frost spells often inflict snare or root effects, as well.
Mages also have a Frost ability category, and obviously any spell in that category is of the Frost school. Of course, Shamans' 
[Frost Shock] is of the Frost school too, as well as the Hunter's 
[Freezing Trap]. Frost resistance exists as a statistic as well. A Warlock's 
[Curse of the Elements] can reduce a target's Fire and Frost resistances.
Shadow
The opposite of Holy magic, Shadow magic is most effective at harming living things.
Shadow damage is dealt by dark spells such as those of the Priest's Shadow Magic ability category and most of the Warlock's repertoire. The 
[Blood Pact] that a Warlock's Imp pet casts to boost the entire party's Stamina is a Shadow effect, as are the creation of a 
[Healthstone] or 
[Soulstone] and the summoning of demons. Most importantly for Warlocks, Enslave Demon is in the Shadow school, and the Warlock 
[Curse of Shadow] reduces the target's Shadow and Arcane resistances. The Priest spell 
[Shadow Protection] increases an ally's Shadow resistance, as can various items.
Holy
The Holy magic School is usually related to abilities which heal and aid living beings. There are also a few spells that do Holy damage: 
[Smite] and 
[Exorcism], for example. Holy magic is especially useful at dispatching the Undead back to their resting places.
Priests and Paladins are the primary users of the Holy school, including all their healing and resurrection abilities.
It is not possible to build resistance to Holy magic, giving Holy Priests an advantage over other casters, but all characters and mobs do have a base resistance chance. Also, the damaging Holy spells used by Priests and Paladins tend to deal smaller amounts of damage in comparison to other schools, counterbalancing the lack of resistance.