According to HowStuffWorks, a hydraulic braking system works by applying force at one point and transmitting it to another point via an incompressible fluid. In a hydraulic braking system for a car, for instance, the force is applied by the driver's foot on the brake pedal. This force is then transmitted to the brakes via the brake fluid.
According to Wikipedia, the first hydraulic brake system was invented in 1914 by Fred Duesenberg. This braking system was first used on racing cars. It was not until 1921 that the technology was used for passenger cars. Unfortunately for Duesenberg, he did not patent his idea, and it is Malcolm Lougheed (also known as Lockheed) who is credited with building a hydraulic brake system for passenger cars.
Wikipedia says that modern cars generally use a power braking system. This type of hydraulic brake system uses a vacuum servo to boost the force that is applied by the driver. These vacuum servos are located between the master brake cylinder and the brake pedal that the driver pushes. Once force is applied to the brake pedal, an air valve opens to allow atmospheric pressure into one of the two chambers of the vacuum servo, creating extra force between the two chambers.