Drilled Pier: A cast-in-place concrete system where a hole is drilled into the ground, the bit removed, a cage of steel dropped in and concrete tremied into the shaft to form a solid concrete foundation. These are sometimes designed with a belled end that is created with a special bit (a can-type apparatus with a swing out plate to form the bell).
Auger-cast pile: These are created by drilling into the ground with a long-stem auger. This is similar in concept to your regular wood drill bit except that it is very long (+30 feet) and has a hollow center where grout can be pressure injected into the bottom of the shaft as the drill bit (hollow-stem auger) is slowly removed. Great when water tables or loose material is present. Steel can be inserted into the top portions of the pile but are not usually extended to the bottom as in the drilled pier.
Caisson: Same as a drilled pier
Pile: Usually the term is referred to a solid piece of material (steel pipe, H shape, precast section, etc) that is driven into the ground with a hammer.
Foundation: A general term used to describe all of the above as well as shallow spread footings.
The best option would be for you to get a copy of a Foundation textbook where the different types are usually described. Try Peck, Hanson, Thornburn, "Foundation Engineeering", published by Wiley.
You might also note that the new IBC 2000 has redefined the difference between a "pile" and a "pier" based on the lengh/diameter ratio and not on any other specific criteria.