A hydraulic lift works by using an incompressible liquid to multiply the effects of the force applied to lift something very large or heavy. A larger surface area requires a smaller amount of force to do the same amount of work.
A hydraulic system generally contains two pistons that are connected by a tube. The pistons and the tubes are filled completely with an incompressible fluid like water or oil. If the pistons have the same diameter, then the force applied on one piston to push it down by a certain distance is the same as the force that lifts the second piston up by the same distance. Since the fluid is incompressible, very little energy is lost to friction.
The relationship between force and the surface area of the piston is defined by the equation Force = Pressure x surface area. The pressure is constant due to the incompressible nature of the fluid. If one of the pistons is built to have a larger surface area than the other piston, then the effect of the force can be multiplied by the same factor as the increase in area of the piston. For example, if a hydraulic system is built where one of the pistons is ten times larger than the other, then moving the smaller piston over 10 meters with a certain amount of force will allow the bigger piston to move by one meter for the same force. Despite the trade off in the distance moved, the force can be used to lift bigger objects.