A header is the collection of tubes that allow the exhaust gases from the combustion chamber to move efficiently into the catalytic converters, the mufflers and out the exhaust tube at the rear of the car.
The differences between manufacturers are based upon the companies technologies and how they "tune" the headers for that specific application.
I owned a 1999 Mustang
GT, and installed JBA headers, which were "shorty" headers. This term refers to the length of the entire header, which is specifically built for that exact year and model of the car. Shorty headers typically allow you to remove the stock headers (which are cast iron, heavy and do not extract the exhaust gases as efficiently as many aftermarket headers) and replace them with their product, also referred to as "bolt on". The engine then can induct more air through the system and in the best case make more horsepower.
However, if the rest of the induction system is not upgraded to allow more air to pass through the engine, the overall affect is not as dramatic. To fully exploit the more efficient headers, the air intake system should be upgraded and the stock catalytic converters and mufflers should be replaced with higher flow devices. This results in a much more responsive engine and creates a much louder car. Replacing just the headers is not enough, but a reasonable start.
Mustang Racing Technology makes an excellent "H-Pipe" that replaces the stock catalytic converter system with a much higher flow device, that bolts onto the stock system. This, coupled with higher flow mufflers (I used Magna Flow) will do more than just the headers and can be done by anyone with a wrench set and a few hours on a weekend. The tone of the exhaust will improve dramatically and you will see an increase in horsepower.
Replacing the headers is not for a first time mechanic on the Mustangs, as you will almost certainly need to spend quite a bit of time under the car and possibly raise the engine off the motor mounts to mount the new headers.