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It's all in the wrist.
Improving your aim in an FPS helps you achieve higher scores with more regularity. Unfortunately, there's a dearth of information accessible to newer players as to how to do that. Most people simply teach themselves how to improve their aim through trial, error and experience, but there are some tricks that you can pick up that will improve your aim by leaps and bounds in only a few sessions of play. These tricks work for just about any game out there, and aren't terribly complicated.
The most important thing to do is to reduce your mouse sensitivity to a very low level - about 1.5-3.0 in most games. This gives you finer motor control over your aim, and protects your mouse from the natural shaking of your hand. Fine control is more important than being able to make rapid turns. In most games, if someone has surprised you from the side, you're likely dead already. You want to be able to pick off their heads easily and quickly, which means being able to steer your mouse precisely without worrying too much about having to overcompensate when you push too far. Cleaning off your table or using a high quality mouse pad if you have an optical mouse will also improve your accuracy.
Putting down your settings until you have a very high frame rate is another trick that competitive gamers use to improve their performance. Turn down your anti-aliasing, shadows, texture filtering, texture quality and High Dynamic Range settings first, and then if you still need to turn things down, reduce the resolution as well. It might make your game look ugly, but you'll be scoring more frags.
Practice aiming at heads by maintaining your aiming cursor at the same vertical level as those of your teammates. For a faster paced game like Team Fortress 2 or Unreal Tournament 3, this trick isn't very helpful, but for games like Counter-Strike Source and Call of Duty 4, it can help you rack up the head shots.
Most realistic and quasi-realistic shooters have the bullets exit your gun in a predetermined fashion. This creates a recoil pattern that can be studied and managed by a savvy player by wiggling your gun in a certain way to follow the path of the recoil and make sure that your bullets always strike their targets. Learn the bullet spread by emptying a clip against a wall, and watching where the bullet holes go for each weapon. Using that knowledge, you'll be a more efficient shooter.