Basically there is a large tank
pressurized with air. All this air is released suddenly and this energy is
what propels the projectile. There are many different designs of pneumatic
cannons. The air can be released by several different methods. A
standard ball valve can be used, a solenoid activated valve, or more complex
diaphragm and check valves.
Standard store
bought valves can have there limitations due to low flow rates. Building
your own oversized check valve can be more effective.
Below is an example of a air
launcher that uses a flexible rubber diaphragm.

Air is filled into the cannon
through the charging hole located on the cleanout cap. The diaphragm is
somewhat flexible and it will bend it's outer edges forward from the pressure
and allow the air to fill the outer chamber. At the same time this forward
pressure is sealing off the end of the barrel preventing air from escaping up
the barrel. Pulling the blowgun out of the charging hole allows the air behind
the diaphragm to escape. At this point all the pressure stored in the outer
chamber bends the diaphragm in the opposite direction sealing off on the outer
edge of the cleanout plug. The diaphragm continues to bend under the pressure
and unseals the barrel end allowing the stored air to escape up the barrel
behind the spud. The diaphragm allows a larger volume of air to enter the
barrel.
Below are some animations and images
that give a better illustration of how this cannon works.
Click to get a bigger version.
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Gun is loaded, with no
pressure. The release valve is shut.
Pressure is pumped in. The
two air chambers equalize, because the pressure leaks around the
diaphragm.
Pressure stays in. The
barrel is a "low-pressure" zone, and sucks the diaphragm toward it. This
keeps it sealed.
Open the release valve,
and the first chamber depressurizes. This sucks the diaphragm away from
the barrel, so the air can escape, firing the bullet.
Gun is empty and
depressurized.
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The flexible diagram material can be
replaced with a piston that actually moves within the launcher. This moving
piston creates a larger opening for the air to get from the air chamber to the
barrel, thereby making the cannon more powerful. A piston can actually move (not
just flex) 2 inches or more. When the air
is charged in the cannon, the piston moves forward and seals against the barrel,
much like the cannon above. When there is a sudden drop of pressure behind the
piston, the piston is sucked back, letting the air enter the barrel.
The piston shown below is constructed out of a 3"
coupler made to fit into 4" pipe.

Check out the P.A.C. 5
A more commonly
known launcher is the "hair spray" potato cannon. This launcher is
basically various pieces of PVC, that are glued together to make something that
looks like a cannon with 2 distinct parts. The first part is the barrel (which
the ammo is shot out of), and the second part is the Combustion/Blast chamber
(where the propellant is ignited). Propellant (hair spray for example) is
sprayed into the Combustion Chamber and the combustion chamber is sealed. The
propellant is ignited and expands, forcing the projectile out of the barrel.

How to use this type
of cannon: First get something to shoot; a potato, apple, etc... Then ram the
projectile down the barrel with something such as a broom handle. Now unscrew
the end cap, spray in some hair spray or other propellant and quickly put the
end cap back on. You're ready!! Now all you have to do is set off the igniter
and BOOM! Off it goes.