As far as "arc" welders go, the stick is the most basic. The user manually feeds the fill metal, which is a stick held in a clamp. Best for large welds on really thick metal or things like cast iron and building ships.
The wire feed welder is an automated version of the stick welder, it uses a spool of flux core wire fed out through the gun at a rate that you set with a dial. Pretty easy to use but too hot for sheet metal and fine work.
The Mig (metal-inert-gas) welder is an evolution of the wire feed welder with the addition of a shielding gas, usually argon and carbon dioxide. It can do much finer welding with minimal spatter but since the feed rate is constant you have to keep moving. Most true MIG welders can be used in either MIG or wire-feed modes by switching the electrical polarity and wire type. Lowes' sells a nice Lincoln setup for about $450.
The Tig (torch-inert-gas) is a process that uses an arc created by a torch with a tungsten electrode that does not erode. A separate piece of filler wire is used, and the operator has complete control over the spark and feed rate of the filler metal with either a foot or hand control. It requires a fair amount of skill to use but a skilled user can make perfect welds in almost any situation.
Most amateurs will probably get the most use and best results from MIG welding, as they are fairly easy to use and purchase.
