Brake horsepower is simply the useful horsepower of an engine or motor. This is how much measured horsepower the engine produces. It is commonly abbreviated as bhp.
Bhp for engines is usually measured with a dynamometer. The engine output shaft is connected to a dynamometer and then a graph is created showing bhp at various rpms. These charts can be misleading, because engines are often tested on stands, instead of in the car. The most valuable measurement is the dyno output at the wheels, as it shows how much power is going to the road after parasitic loss.
I don't remember where I picked up this fact, but BMW aims for 100 bhp per liter of displacement when designing their engines. The 3 liter M3 engine makes 320 bhp.
Brake horsepower is very important to airplane engines and turbines. when airplanes or helicopters are operating with one engine inoperative, the other engines must be able to operate at such a performance as to land the craft safely. There are a few bhp ratings for aircraft engines.
A common Rolls-Royce helicopter turbine makes 420 bhp at max, but can only do that for 30 min before the turbine is damaged. It normally outputs between 275 and 333 bhp, while cruising. Rolls based these brake horsepower calculations on lab and destructive testing of the engines with a dynamometer attached, then added a safety factor.
FAA and Jeppsen-Sandersen literature