Too many have been in existence to be feasibly named here. If you narrow down your criteria, perhaps even include which era you had in mind, a more specific answer may be given. Some weapons, both past and present, which match your criteria include (but are not limited to)...
Both air cooled and water cooled machine guns have been made.
Liquid cooled. The last "sport" bike to be air cooled was the WAY out of date Katana
water cooled is better
Almost all of the first aircraft engines were air cooled as this made for a lighter engine.
The machine gun evolved from the Gattling gun, a device that that rapidly fired bullets when cranked. WWI saw the first true machine guns, liquid and air cooled. http://en.wikipedia.org/wiki/Gatling_gun Another example is the french Mitralleuse, used in the Franco/Prussian war of 1870/71.
Lewis gun
The Kalishnikov PKM is a gas operated, belt fed, air cooled, automatic only, machine gun.
Yes- belt fed, air cooled, gas operated machinegun. Gas operated means it uses some of the expanding gasses from a fired cartridge to operate the action.
The M240B is an American variant of the Belgian FN-MAG machine gun. It is a belt-fed, gas operated, air cooled machine gun which fires the 7.62x51 NATO cartridge.
Vickers- belt fed water cooled medium machinegun. Vickers a light, air cooled, magazine fed.
The M249 is an automatic light machine gun. This gun uses 5.56 mm ammunition. It is gas operated and air cooled. It has a quick change barrel. It can support a bi-pod attachment, allowing a boost in the already accurate rate of fire the gun possess.
Both air cooled and water cooled machine guns have been made.
Early machine guns such as Maxim and Vickers were water cooled and had distinctive jackets around their barrels, except models intended for installation on airplanes. Most later machine guns are air cooled to make them lighter and more portable.
The chiller is machine that evaporates liquids through an absorption process. This machine also cooled air and equipement. In others words this machine is and Air-Water Cooling System.
By the development and manufacture of lighter, air cooled machine guns.
Air cooled, magazine fed, shoulder fired, gas operated, lightweight.
Browning m1917 water cooled browning m1919 air cooled browning m2 .50 caliber machine gun