They make use of engine vacuum on one side of a diaphragm, and atmospheric pressure on the other side. A valve allows the vacuum to act on the diaphragm in proportion to brake pedal travel, while a check valve prevents vacuum from bleeding back into the engine. This assists the pedal effort, and allows increased pressure on the brake fluid without a corresponding increase in pedal effort. Other types of boosters use hydraulic pressure, either from the vehicle's power steering pump or from a separate electric pump, or both, to assist pedal effort. As the brake pedal is depressed, a valve increases hydraulic pressure in a boost chamber to apply increased pressure to the master cylinder pistons.
Some systems use both vacuum and hydraulic assist.
In other systems, air pressure, from an on-board compressor, is used to generate hydraulic system pressure. Modern hydraulic brake systems are split systems, comprising two discreet braking circuits. One master-cylinder piston and reservoir are used to actuate the brakes on one axle, with a separate piston and reservoir actuating the brakes on the other. If a leak develops in one hydraulic circuit, the other will stop the vehicle. Once stopped, of course, the vehicle shouldn't be driven until the brake system is repaired.