Pneumatics is the field of using air to cause mechanical movement. In practice, it is a great way to automate the movement of props and displays. Motors can be used to make things move too, but pneumatics is useful for a broader range of movements, both fast or slow, big or small.
The core of pneumatics is the pneumatic cylinder. The cylinder consists of a tube with a metal rod inside. The cylinder has one or more connections for the flow of air into and out of the internal air chamber. When air is forced into the chamber, it pushes the metal rod outward. The linear movement of that rod is used for mechanical movement. When the air pressure is reduced, the rod can retract, or air can be forced into the other end of the cylinder to apply force and return the rod to the retracted position.
The second core component is the pneumatic solenoid valve. Pneumatic solenoid valves are electrically activated to stop and start the flow of air, similar to a faucet stopping and starting the flow of water. These valves can offer more features than just "on" or "off" but we'll get into that later.
The final core component is air supply, typically provided via an air compressor. In a pinch you can use an air tank pressurized for free at the local service station. Additional components used in pneumatics include tubing or pipe, triggers, air filtration, flow controls, mufflers and a variety of other accessories.
Pneumatics has the useful attributes of being able to provide fast or slow movement that is infinitely repeatable. This is ideal for shows, whether an elaborate holiday display, a walk through haunted house or stage props. Furthermore, it can be set up to operate on a timed schedule, to be triggered by an event (such as a person stepping on a trigger mat or in front of a motion detector) or it can be triggered manually.
Pneumatics is essentially an industrial process and so there isn't very much home consumer information available. However, pneumatics isn't terribly complicated, especially because the requirements of prop animation usually don't have the same strict requirements that industrial processes often have. The connection of cylinders to air valves to air supply is pretty modular. Purchase components that all use the same connector size and everything goes together almost tinkertoy-like.
Read about each component to learn more about how to select the right equipment for your purposes and how it works in more detail. After that check out some examples and creative suggestions for using pneumatics in your project.