Keep a 'Baggie' over the end of the burp gun to minimize the damage that chemical fumes and ozone will do to the rubber ring on the end of the gun.  To operate the gun just bring the two handles together sharply which will result in a large increase in air pressure inside the gun.  Eventually the air pressure increase will overcome the friction of the rubber ring holding the ball in the gun allowing it to "pop" out.

The potato gun will shoot its "projectile" by a compressed air type mechanism just like the burp gun described above.   Press the muzzle of the gun into a potato and the twist out.  Compressing the handle of the gun will increase the air pressure inside the gun until the potato projectile pops loose.