A reconstruction of the Wright Brothers' airplane catapult, sponsored by the Ohio Bicentennial commission and The Ohio Historical Society
In their efforts to create a flying vehicle, the Wright Brothers created a catapult to launch early versions. It wasn't a catapult in the Game of Thrones sense, but rather operated by constructing a tall framework tower, one that looked like an oil derrick, and dropping a huge weight down the center of it. The weight was attached to a rope and ran down a track that the aircraft was mounted on; at the nose end of the track, the rope did a 180 around a pulley, coming back to attach to the plane; so when the weight was dropped, the plane was effectively yanked forward to get it going.
The video below is of poor quality—it's from 1909, so it's a miracle we can see it at all—but if you fast-forward to about 3:19, you can see the derrick/catapult and subsequent launch of an early Army test plane, piloted by Orville Wright himself:
Once airplanes became established vehicles and the military decided they had value, it was just a matter of time before the technology advanced. Tri-planes, bi-planes and later fixed-wing aircraft had engines powerful enough to launch them of their own accord, but a later 20th Century invention—the aircraft carrier—had runways of limited length. To get around this, the British military devised, in the 1950s, an ingenious steam-driven catapult to launch aircraft from a shorter track. Here's some test footage from the era:
British engineers worked out the kinks, and the technology soon spread. The way it works is that steam boilers in the ship build up and store a tremendous amount of pressure. A channel in the runway deck contains a piston-like shuttle that is attached to the plane's front landing gear.
The steam pressure is built up behind the shuttle. When it's Go Time, the shuttle is released and shoots forward, dragging the airplane with it. The airplane works up enough speed to leave the deck, and then the now-detached shuttle runs into a channel of fluid—"water brakes"—to slow it down.
