MASTER CYLINDER
The master cylinder is located in the engine compartment and is typically actually two cylinders in one housing, each connected to two wheels. In this way if one side fails, you will still be able to stop the car. The master cylinder should be checked regularly to ensure the correct level of brake fluid is in the system. If the fluid is low the entire system should be checked for leaks.
VACUUM PUMP (BRAKE POWER BOOSTER)
The vacuum pump is usually mounted on the master cylinder and is directly connected to the foot pedal. Its purpose is to amplify the foot pressure applied to the brake pedal by drawing power from a vacuum created by the engine. The pump will only work whilst the engine is running which is why your brakes do not work so well with the engine off. To test the pump, put your foot hard on the brake pedal with the engine off, then turn on the engine and the pedal should depress slightly.