An arc welder is also called a stick welder. They come in ac or dc, and some are switchable, ac and dc. It uses flux coated rods and is best for welding mild steel. The flux coating produces a shielding gas to keep oxygen, nitrogen and hydrogen from the atmosphere from contacting the puddle and forming occlusions (pits and voids) which weaken the weld. Flux coated rods are made for other alloys, such as stainless steel and aluminum, but they are difficult to use and produce an inferior weld in these metals.
MIG stands for metal inert gas, and the welder doesn't use rods but feeds a spool of wire from the wire feed electrode or "MIG gun". The wire is solid metal (no flux) because inert gas, usually CO2 or Argon, is supplied from a bottle and fed through ports in the gun to shield the molten puddle. With different alloys of wire, you can weld many types of metal with ease.
Professional welders almost universally prefer MIG welders to stick welders because they are more versatile. A MIG gun can get into a tighter space than the "stinger" of a stick welder. Once you get used to wire feed, which takes some practice, it is easier to control and maintain a steady arc. MIG welding is a little more difficult to use outdoors, especially in windy conditions, where the wind can blow away the gas. So wind breaks are necessary. Many people state that you also don't get as deep of penetration with MIG compared to stick. I've talked to welders who counter that statement and say you just need a bigger machine (higher amps) and you can get any penetration you need with MIG - and I don't know which opinion is correct. The MIG welders I've used were small machines (125 amp) whereas most of the stick welding machines I've used were rated 250 amps. I could definitely get more penetration and weld thicker stock with the latter, but it could just be the fact that it was a much larger machine.