In an engine, air and fuel is mixed together into the cylinder. The spark plug ignites the mixture, and the exhaust fumes are pushed out into the header. From there, it is led to the catalytic converter (an anti-pollution device). It then goes through the exhaust system.

The purpose of getting an exhaust system is so that it can let the exhaust fumes get out of your car faster (instead of being bottle necked). Next time you're in a McDonald's, try blowing through a coffee stirrer/straw, and then try to blow through a regular straw. You'll find that it's easier to blow air out of a straw with a larger diameter.

So why not just drop the exhaust system so that there isn't a bottleneck anymore? There are two reasons...first, it allows the exhaust system to muffle the exhaust tones (hence the term muffler...it muffles the sound that travels through the pipe from the engine). The second is that a little bottleneck is good...it gives a little backpressure which the engine needs to work effectively.

Aftermarket exhaust systems are usually louder because they are less restrictive. In a stock system, the exhaust fumes in the muffler have a longer distance to travel since the pipe inside goes back and forth. It needs this longer distance so that the exhaust has to go through more of the muffler's packing which absorbs the sound. In an aftermarket system, the exhaust isn't led on a longer path...so not as much of the sound can be absorbed.

With an aftermarket exhaust system, and an aftermarket intake/header, you'll be able to have more air flowing through your system...this allows your engine to make more power at higher rpm's. For example, sucking air through a straw slowly won't be a problem for you (you can breathe through it while sitting down). But if you ran around the block as fast as possible, then put a straw in your mouth...you'll probably have a hard time breathing. Thus, when the engine is at higher rpm's, it needs to be able to breathe easier (intake)...but it needs an exhaust system to escort the previously used air out...