Hydraulic power works by applying force at one point to transfer it to another point using an incompressible fluid. The incompressible fluid is usually an oil of some kind.
Hydraulic power is almost always used to multiply force. Hydraulic multiplication is as easy as changing the diameter of the cylinder and piston to be smaller than the others. If the larger piston is five times the diameter of the smaller piston, the smaller piston moves five times as much as the larger piston and applies five times as much force, similar to a pulley system. So, if 100 pounds of force is applied to the small piston, you get 500 pounds of force from the large piston.
The brakes on a vehicle are a good example of a hydraulic system. When the brake pedal on a vehicle is pressed, it pushes on a piston in the master cylinder, which then transmits that force to the pistons at all four wheels. The pistons at the four wheels push the brake pads against the rotors, slowing the car down. If it wasn't for hydraulics, it would be impossible to apply enough pressure to stop a vehicle. Hydraulics do not function optimally, however, if there is air in the fluid lines.