MIG stands for Metal Inert Gas. It is the popular name for the GMAW (Gas Metal Arc Welding) process. The name MIG is technically incorrect because some of the shielding gases used are not inert. It is also called wire feed welding. The other process that uses the same configuration is FCAW (Flux Core Arc Welding). Any welder which will run wire with gas shielding will also run flux cored wire without. Some inexpensive welders are made to run flux cored wire only, they do not have the capability to connect and run shielding gas. You can run aluminum wire with a wire feed welder and the right shielding gas. It works best if you have a piece of equipment called a spool gun which looks like a hair dryer and attaches to the business end of the hose. This is because aluminum is softer than steel and may not feed well when pushed too far.
TIG stands for Tungsten Inert Gas. It is the popular name for GTAW (Gas Tungsten Arc Welding). TIG welding uses a non-consumable tungsten electrode, and the filler material (if required) is supplied by hand, as with oxy-acetylene welding. With TIG, you use whichever shielding gas is appropriate to the type of metal being welded. TIG is generally considered the best process for aluminum welding because it is clean and precise. It is also very slow. It would not be considered the best process for welding in an industrial setting making boats, for example. It would be too slow to be cost effective.
Good quality TIG welders are very expensive, largely because of the electronics involved in having so much control over the characteristics of the arc.