What is the difference between hp and bhp?

Horsepower and Brake horsepower are really the same thing as far as mathematical calculation. The difference is how it is measured. Brake horsepower is usually a measure of an engine's horsepower at the shaft WITHOUT anything attached to the engine that would normally be required like the water pump, alternator, power-steering pump, airconditioner, etc. Brake horsepower is usually a theoretical calculation or a measurement made only in a lab under controlled conditions since an engine cannot run without most of the things that it would be required to remove. In the lab/shop they would conect the engine to the dynomometer and then supply coolent, oil pressure, electricity, etc through external means.