You're probably familiar with the idea of spells in the Word of Warcraft. These are powerful abilities whose use takes a few seconds to complete. This represents "casting time." If a character gets through the entire casting time for a spell, the unleashed ability will do significant damage to its target (or, in the case of heals, will restore a significant number of hit points). Whatever the purpose of the spell, when the casting time is complete, big things happen.
It is therefore attractive to opponents to stop the spell from casting. Sure, killing the boss will keep the spell from finishing. But that's far from the only way to stop a spell's completion. Abilities that stop a spell's cast are called interrupts. This is because they interrupt a spell cast. (See? It's not just a clever name.)
The basic purpose of interrupt abilities is to stop a spell from being cast by the target. These abilities, like Pummel or Kick, stop a spell instantly and often prevent other spells from being cost for a short time. They don't do any damage or apply any crowd control; they just stop a spell from being cast.
These abilities are frequently on a very short cooldown. They are intended to be used frequently and be available when needed. Which the low resource cost and high availability, there isn't much reason not to use interrupts when you can.
As a general rule, you should nearly consider interrupts "the new DPS." If you're going to spam a meter at the end of an instance, go with interrupts over simple DPS. Interrupts are an important utility your character possesses for a reason.
Of course, there are other ways to interrupt a spellcast. Most abilities that apply a crowd control effect -- like Polymorph -- can be used in a pinch. Sure, the CC will break momentarily, but the spell has been interrupted. Polymorph isn't actually a great example for use this way, because of its long cast time. But the idea is that by briefly crowd-controlling the target, you've robbed them of the spell.
The other way to interrupt spellcasting is to force the other character to move. Knockback spells like Typhoon often frequently interrupt spellcasting, as they move the target. Of course, not all boss mobs are able to be knocked back, but it's worth a shot.
It's not enough to simply interrupt the spell. You should have an idea of why you're doing it; understand what's happening under the hood will help you use your interrupts more wisely.
First, mobs that use spells tend to hit harder than melee mobs; this isn't a universal rule, of course, but it's generally the case. Because of that, you can interrupt the NPC's spells to mitigate the amount of damage being done to the tank and that the healer must handle after the face.
Second, not all spells simply do damage. NPCs use crowd control and buffs, as well. If the NPC spell effect has a casting bar, then you absolutely want to interrupt it. After all, if your enemy wants something to happen, then you probably don't want it to happen.
Third, in Cataclysm, the tanking and healing game is different. The healer is put under immense pressure to keep the group alive. Everything you can do that takes pressure off the healer is a huge benefit to the party.
Last, many boss mobs are built with abilities that simply must be interrupted. The idea is that you either interrupt the spell or the party will wipe. Heroic Throne of Tides features Lady Naz'jar's Shock Blast, for example. Sure, with a strong enough tank, you could live through it. But it does such a huge amount of damage that you need to stop it if you can.
How to practice
I feel like I say this all the time, but if you want to practice your ability to interrupt enemy spells, get yourself into some PvP. Because player opponents cast spells unpredictably, it's the best way to hone your skills at quickly reacting to enemy actions. Additionally, PvP is so busy that you'll have to learn how to always be on the lookout for spells even while focused on other tasks. That's a great asset to have and will go a long way toward helping you make your interrupts more effective.
Visit the WoW Rookie Guide for links to everything you need to get started as a new player, from how to control your character and camera angles when you're just starting out, to pulling together enough cash for mid-level expenses such as mounts, to dungeoneering and travel tips for lowbies.