This came up recently, so I figured I'd spell it out for all of you. Anyone with a magic stat can attempt to counter a spell. Having practice with spells in general, knowing what spell is being cast, and knowing how to cast that spell yourself, all make countering the spell easier. 

Game mechanics wise, counter-spelling is an opposed Magic roll. You have to notice the mage is casting something (with a Perception check) in order to do something about it soon enough. The possible results are the spell is countered (has no effect, energies dissipate harmlessly), disrupted (spell doesn't have normal effect, energies are released and can cause other problems), weakened (the casting roll for the spell is harder, and its effects less impressive), or no effect (your feeble attempts don't even impede the other caster). 

Fluff wise, you are disrupting the flows of energy the other caster is building, interrupting their focus and generally making a mess of the nicely ordered spell they are creating. This is largely a mental exercise, requiring a lot of concentration and does tire you out (similar to casting a spell, although generally not as badly). Hand gestures are common as are fancy light shows, especially if the spell gets disrupted. Shouting or other distractions can also help.

Non-magic users can attempt their own form of counter spelling. It usually involves introducing a sharp piece of metal to the spellcaster's body. The resulting shock and pain tend to make spellcasting considerably harder.