Submachine Gun: Magazine fed, select fire weapon firing pistol ammunition. Example: Heckler & Koch MP-5 series.
Assault Rifle: Magazine fed, select fire weapon firing intermediate rifle ammunition. Example: Colt M-16.
Assault Carbine: A more compact form of assault rifle, generally equipped with folding stock. Example: Colt M-4.
Light Machine Gun / Squad Automatic Weapon: Belt-fed, fully automatic weapon firing intermediate rifle ammunition (preferably, fires same round as assault rifles in order to simplify field logistics). Generally used with built-in bipod, but can be mounted if needed. Example: M-249 SAW, a.k.a. FN Minimi.
Medium Machine Gun / General Purpose Machine Gun: Belt-fed, fully automatic weapon firing full-power rifle ammunition (7.62x51 NATO). Generally fired from tripod, but sufficiently light to be used with bipod as an infantry-carried weapon if need be. Example: M-240, a.k.a. FN MAG.
Heavy Machine Gun: Belt-fed, fully automatic weapon firing large-caliber rifle ammunition (.50-cal, 12.7mm, 14.7mm, etc). Too heavy to be fired without tripod mounting. Example: Browning M-2HB.
Gatling Gun: Belt-fed, fully automatic weapon with multiple rotating barrels. Can be in pretty much any caliber (U.S. has 5.56mm, 7.62mm, 20mm, and 30mm), but a heavy weapon regardless of caliber. Excessively high rate of fire (10000 RPM for 5.56mm Microgun, 6000 RPM for 7.62mm Minigun, 6000 RPM for 20mm M61 Vulcan, 3900 RPM for 30mm GAU-8 Avenger). A rather specialized class, but also very useful.