Normally, you can counter a spell three ways. For all of these, you need to have readied an action to counterspell:
1. You identify the spell being cast (Spellcraft check DC 15 + spell level), and you happen to be able to cast the same spell, or a spell that specifically counters it, like slow vs. haste. You cast the spell as a counterspell, and unless your spell is interrupted somehow, you counter your opponent's spell.
2. As above, except that you have the Improved Counterspell feat, and instead of using the exact same spell, you use a spell of the same school but of a higher level. For instance, you could use a shatter spell to counter magic missile, since both are evocation, and shatter is higher level than magic missile.
3. You cast dispel magic, targeting the opposing spellcaster. Even if you don't know what spell is being cast, you roll d20+your caster level vs. DC 11 + your opponent's caster level. If you succeed, you counter the spell.
Normally, almost no one uses counterspelling because it's a crappy tactic. You're almost always better off just readying to cast magic missile at your opponent whenever you starts casting. You deal damage, and though your opponent gets a Concentration check to keep his spell, you at least have a chance to come out ahead (i.e., for the cost of your spell, you negate his spell and deal some damage). With normal counterspelling you just trade one of your spells for one of his, so you negate your opponent's tempo, but this is only useful if you have more allies than he does.
Inquisitors have two abilities that make counterspelling more appealing.
First, the reactive counterspell feat lets them counter even if they haven't readied an action. This way, if your opponent doesn't cast a spell, you haven't wasted your turn readying an action that never goes off. If he does cast a spell, you can counter it, and you lose your standard action on the next turn.
Second, inquisitors trade the normal clerical ability to turn undead (or in their case, rebuke/command/bolster undead, because they're evil) to instead rebuke magic. Normally, you don't get access to dispel magic until 5th level, but inquisitors get to do much the same thing as early as 1st level. They can channel negative energy to counter a spell, and though they must succeed a check to do so, they don't have to know the same spell.
Counterspelling is the inquisitors' schtick, and whenever they show up in the campaign saga, they always have minions, and usually are higher level than the heroes so they will be a credible threat.
Does that clear things up?
