Popguns don't have to be powered by air pressure. Evil Genius Dr. Simon Monk has designed lots of cool electronics projects, including a coil gun that uses magnetic repulsion instead of air pressure.
I have fond memories of a wooden popgun I had when I was a little kid. It helped me begin to figure out what air pressure was all about.
Popguns are cousins to reciprocating air and water pumps, as well as piston-type engines. In all cases you have a cylinder with a high pressure side and a low pressure side. Whatever slides back and forth in the cylinder has to form a seal with it so pressure doesn't leak past.
In gasoline engines the task is accomplished with at least two rings which fit into machined groves in the piston. They spring out gently against the sides of the cylinder, as indicated by green in the cutaway illustration (the third purple ring is for oil). When the rings get worn out, the engine becomes weak and inefficient because some of the pressure blows by the rings instead of pushing on the piston.
For air and water pumps a cup-shaped piece of leather (red in the cutaway illustration) often makes the seal. It cleverly uses high-pressure air or water to push the leather against the cylinder wall. The higher the pressure that must be contained, the harder the leather seal is pressed.
The "POP" of a popgun comes from the sudden release of air pressure, which causes a shock wave.