A hydraulic jack consists of two connected cylinders, one big and one small, where force is applied to the smaller cylinder that creates equal pressure in both cylinders. Since the large cylinder has a larger area, the large cylinder produces greater force. By applying force to the small cylinder, the force is amplified in the bigger cylinder.
Hydraulic jacks work based off of Pascal's Principle, which states that pressure is equal throughout an enclosed container. When force is applied to the small cylinder, the pressure in the cylinder gets greater. By Pascal's Principle, this means pressure in the larger cylinder gets greater as well. But, due to its size, the larger cylinder produces a greater force than the smaller one, allowing for multiplied hydraulic force.