This is a classic HowStuffWorks article that describes the construction and operation of the traditional potato cannon:
The traditional potato cannon is a combustion device – it uses hairspray as its fuel and the spark unit from an outdoor grill to ignite it. This works fine but it is a little fidgety to get it working. You also have to open the back of the gun up for every shot in order to spray in more fuel, and too much or too little fuel can cause the gun to misfire.
The new design shown in the following video uses pressurized air instead, making it simpler to build and to operate the gun: