MIG, which stands for "Metal/Inert Gas", is a type of welding using an automatically fed wire. This is also called "wire feed welding" and "semi-automatic welding" because rather than using welding rod, which you control the rate of manually by applying it at different speeds and pressures, the welder feeds a wire at a given rate. The wire is usually steel, and the gas is usually Argon. Many people use carbon dioxide (which is not inert) because the oxygen in it is actually beneficial when used on dirty surfaces, and it is much cheaper than Argon. The gas is used to keep oxygen from coming in contact with the metal, because at high temperatures, steel will burn (oxidize) when it comes in contact with oxygen. This causes the metal to oxidize and the welding wire to spatter.
In fact all wire-feed welders are referred to as MIG welders, even when no gas is being used. Gasless MIG requires the use of flux-core wire (like flux-core solder) which cleans the surface as it oxidizes. Unfortunately, that causes a great deal of spatter, and the flux generally makes a mess which must be cleaned up later. Carbon dioxide is less than perfectly clean as well, and the best looking welds come from using a genuinely inert gas. Argon is generally the cheapest and most available of these.
MIG is lower-power than other forms of welding, and as such can be used on thinner metal such as auto body than higher-power arc welding like TIG (Tungsten/Inert Gas) or straight traditional "arc welding" done without gas. It uses far less heat than oxyacetylene welding as well. It is easy, because the wire is fed at a given speed, so all one has to do to MIG weld is to get a feel for it (you can get the basics in about half an hour) and then be consistent. A decent MIG welder will run you about $350 without gas, or $600 with gas; Gas is of course what makes them really useful and allows you to lay down clean, nice-looking welds, so it's worth the additional expenditure.