Chain guns are a type of automatic weapon. The only reason its called a chain gun is that it uses a cam and chain to perform the feed and extraction functions (I'm not sure about the chambering function). Often this is further sped up by using a rotating breach, that rotates so that the mechanism is set up that every time the gun fires, a round is being fed into the system and extracted at the exact same time. Unless I am mistaken almost all chain driven weapons are single barrelled. In a rotary barrled weapon, the breach often rotates counter to the barrels to prevent destructive torque in the system. A further note on rotary barrled weapons is that there is only ever one breach and one firing barrel in use when the weapon fires, and its simply because of the speed at which the system can be 'cranked' that is why one can achieve the ridiculous ROFs.
Chain guns are externally powered by an electric motor, which drives the entire firing cycle by means of a chain. This means that the rate of fire can be easily varied by adjusting the electric power, and the weapon is very reliable as the gun function doesn't depend on every cartridge firing properly (although that isn't such an issue today given modern cartridge quality). Chain guns usually have low to moderate rates of fire, and only a single barrel. They are available in calibres from 7.62mm to 35mm (with 40mm and 50mm on the way).
Other types of externally-powered guns are the multiple-barrel American rotaries, in which the entire cluster of barrels rotates as the cartridges are chambered and fired. Each barrel has its own chamber, breech, bolt and firing mechanism which rotate around with the barrels (not in the opposite direction). The number of barrels varies from three to seven, and calibres in service range from 7.62mm to 30mm. They usually have a high rate of fire.
Most other guns (and most Russian rotaries) are powered by the cartridges so they don't need an external power source. They may be gas-operated (using gas tapped from the barrel) or recoil-operated (using the recoil of the barrel to drive the reloading action).
Tony Williams: Military gun and ammunition website and Discussion forum