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 line of the spell) and Warlocks (if it doesn't involve fire, it's Shadow). For other classes it's not always clear.
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.
Arcane
Arcane magic typically inflicts direct damage.
Obviously any spell from a Mage's Arcane ability category is of the Arcane school, including both spells that do damage ( [Arcane Missiles], [Arcane Explosion]) and spells that don't ( [Polymorph], [Arcane Intellect]). But Mages aren't the only class; Hunters have [Arcane Shot], and Druids' [Moonfire] and [Starfire] spells do Arcane damage. Hostile Arcane spells can be resisted; there are items that increase one's Arcane resistance. Night Elf Priests can also do Arcane damage, with [Starshards].
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. A Shaman's [Flame Shock], a Mage's [Fireball], and a Warlock's [Immolate] are all quite capable of causing damage in this school, but a mage's [Fire Ward] and a shaman's [Frost Resistance Totem] are non-damaging abilities in the Fire school. Like Arcane resistance, Fire resistance is a statistic that can be affected by spells and items.
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. 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 [Holy Nova], 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.
Holy magic apparently doesn't have a resistance indicator in character information, but the game does track it. Case in point: Holy Protection Potion.