Atomizers, essentially, are the engines of
vaping with no fuel tank attached; they are fed either by a cartridge (a foam-filled container which holds
juice and delivers it to the atomizer coil by capillary action) or by direct dripping, in which
juice is dropped by the user directly into the atomizer.
Cartomizers are really atomizers with an integral chamber packed with filler material, which delivers juice to the coil in much the same manner as cartridges.
As a general rule, atomizers fed by direct dripping produce the best, most saturated flavor, but you have re-drip after a few draws; cartomizers last considerably longer without refueling, but flavor is compromised in order to gain convenience.
Cartridges, IMHO, are the least-desirable option.