In this demonstration, students observe a very simple hydraulic system at work: the use of water to lift a person.
The pressure at the bottom of a column of water depends on the depth of the water. Imagine diving into a swimming pool. As you dive deeper, there's more water pushing down on you, so the pressure on you is greater.
Pressure is force per unit area. The pressure at the bottom of a container doesn't depend on the shape or volume of the container, only on the depth of the water. The pressure at the bottom of the long hose in this experiment is large, because of the depth of the water above it.
The hot water bottle has a much bigger area than the opening of the hose. Since pressure is force per unit area, the same pressure on a much bigger area creates a big force that can lift the person standing on the hot water bottle.
When you push on the smaller piston (on the left) you create pressure in the liquid. The same pressure on a much bigger piston exerts a much greater force, so the second piston could lift a car or an elephant!
A system like this, where a fluid is used to transfer force, is called a hydraulic system. Hydraulic systems are used to lift heavy objects.