Well the 3 basic ones are Electric, Gas and Spring.
Electric ones are powered by a motor & piston and run on a battery. There are different kinds of batteries (Nickel Cadmium - NiCd, Nickel Metal Hydride - NiMH, Lithium Polymer - LiPo are the three common ones) and come in different voltages (7.2V, 8.4V, 9.6V are common for Nickel based ones, 7.4V, 11.1V for LiPos).
LiPos are generally better in terms of charge capacity and such, but need special balancing chargers.
Usually, the higher the voltage, the higher the rate of fire of the electric gun.
The higher the capacity (in mAh), the longer the battery lasts before needing a recharge.
Then there are gas guns, which are powered by either propane (most common, also called green gas, top gas or HFC22), CO2 or duster gas (also called HFC134a).
Duster is usually easy on the internals of the gun, but low power. CO2 is the opposite of that. Propane is the 'just-right' one in some ways.
Gas guns come in rifles and pistols. Both generally hold the gas in the magazine of the gun, along with BBs.
Last are springers. Not very useful in pistols or assault rifles, but there are bolt action sniper rifles that are spring based and very effective. You effectively compress the spring when you pull the bolt back and cock the rifle, and then fire.
H&K G36c (KWA) / Glock 18c (TM) / Kimber LAPD SWAT Custom (TM) / Mossberg M500 (ACM)