Headers are basically the pipes that connect to your engine/cylinders for exhaust expulsion. If you have a 4 cylinder car, you have a header with 4 pipes going into one tailpipe. If you have a six inline, you usually have six going into one pipe.
The faster you can expell the exhaust, the more horsepower you get from your engine since it has to work less to "throw out the trash". The larger the header, the faster the gases are released. If you have a larger header and it goes into a small tailpipe, it does you no good. You have to make sure that if you want to spend the money on a bigger header, you also should invest in a larger pipe system for the entire exhaust of the car.