There are many different designs of hydraulic car lifts in use today. The most common designs have two or four posts with arms that extend beneath the car to lift the car by the frame (or at specified jacking points). Another common design, sometimes called a "drive-on" lift, has solid metal tracks for the car to drive onto before being lifted. Other designs sometimes use a centre column sunk into the floor beneath the lift. Regardless of the design, all car lifts operate using hydraulic systems.
A car lift operates using the same basic concept as any hydraulic system: when you apply force to a liquid in one place, the pressure is transmitted through the system to exert an effect somewhere else. Car lifts use hydraulic fluid (a petroleum oil with additives), which cannot be compressed no matter how much pressure you exert on it. Instead, the fluid flows through the hydraulic system and moves a cylinder that raises the car off the ground. Some car lifts use an air compressor, while others use an electric motor. Regardless the power source utilised, the basic idea is the same. The lift exerts force on the hydraulic fluid, which in turn moves a cylinder to raise the car. Trading force for distance is a common idea in mechanical systems. In the case of a car lift, this means connecting a narrow cylinder to a wide one via hydraulic lines. Oil compressed through the narrow cylinder travels a great distance. When that force is transferred into the wide cylinder, it moves a shorter distance, but with much greater force.
The same theory is used to operate the brakes on your car: when you press down on the brake pedal, the force is transferred through the fluid in the brake lines to the master cylinder, which transmits force equally to the calipers at all four wheels. Because the brake fluid cannot be compressed, you can push down harder on the pedal to exert more pressure on the brakes. (This is why it's crucial to eliminate any air bubbles from your brake system by "bleeding" the brake cylinders. Air, unlike hydraulic fluid, can be compressed. So if air bubbles are present inside your brake lines, the force you apply to the brake pedal is used in part to compress the air bubbles instead of pushing on the calipers, which reduces the efficiency of your brake system.)