AIAD: Basically an internship similar to what you might see at a civilian college, but with some fundamental differences. Examples: DOD, White House, State Department, FBI, Corporations, Banks, Stock Exchange, Research Labs, etc. You apply through USMA rather than directly to the source. USMA gets the slots and you compete with other cadets for a spot. Selections are made by USMA Officers/Instructors. Available when you meet qualifications and can fit into schedule. Can be done after Plebe year, but schedule restrictions and prerequisite courses make it more difficult.
MIAD: Military training - Airborne, Air Assault, Sapper, etc. Most cadets do Air Assault after Plebe year. Some do Airborne. Beyond that you apply and compete for other schools depending on schedule and required qualifications.
PIAD? Not familiar with that one.