There are a few reasons a cold air intake is a popular modification for someone who's just starting to modify a car. Cold air intakes generally cost just a few hundred dollars (depending, of course, on the manufacturer of the intake itself and the type of car you own). They're also relatively easy to install with just a few hand tools in an average-equipped garage. And despite their simplicity, a cold air intake actually provides a decent horsepower boost for a small investment.
A cold air intake works by moving the engine's air filter away from the engine, so that the air being sucked in is cooler. Normally, air is drawn from the area right next to the engine, which, of course, is very hot because the engine heats up everything under the hood and all around it. Cooler air carries more oxygen, so if you can help your car somehow reach air that hasn't been heated by the engine, that extra oxygen makes a bigger bang from the same amount of fuel. That might not sound like such a big deal, but think about exercising outdoors on a hot and humid day, as opposed to a day with a cool refreshing breeze. Unless you're an elite athlete, no one's diligently measuring how the weather affects your body's performance. But the difference in the way you feel has a very scientific basis, and the same basic principle applies to your car. That's why weather reports give smog warnings on gross, congested days -- hot, dirty air leads to all kinds of ailments.
Cold air intakes are designed to flow more efficiently than your car's from-the-factory, traditional airflow assembly. Stock tubes are often small and restricted; an aftermarket intake has larger diameter, smooth tubes to encourage stronger airflow. That doesn't mean that factory-installed systems are completely useless -- all that "extra" filtration is designed to help your car stay clean under all circumstances. But if you want extra power, you have to be willing to compromise a little in the hopes of attaining optimal conditions instead of average conditions.
If you want your car to run better, opening up its airflow is a good start. That's why a cold air intake is a small investment with a rather big payoff. It makes such a big difference, in fact, that the simple process of redirecting the filter to draw cooler air is good for a horsepower gain of about 5 to 20 ponies in most cars. It might even improve your fuel efficiency, and it'll probably make your engine sound better, too. And the standard disclaimer, of course, is that your exact results will depend on the specific make and model of your car and the cold air intake system you choose.