The AR-15 is the most commercially successful of the AR family. The M-16 is simply a select-fire version of the AR-15, as is the M-4.
The AR-10 is a 308 rifle based off of Eugene Stoner's earlier design.
Unrelated except in name, the AR-180 used a stamped receiver and shared no parts with the AR-15 as originally shipped. However, it seems that AR-15 FCG parts could in fact be used in it, just not in a configuration which resembled the original Armalite FCG. It also did not accept STANAG magazines. The AR-180 and AR-180B are usually what cause folks to exclaim that they've seen an AR-15 upper with a side-mounted charging handle. (Although American Spirit Arms does produce an AR-15 upper with a left-side charging handle)
The AR-180B (newer rifle) apparently shares more parts with the AR-15, and does accept STANAG magazines.
Generic Description of the AR Platform
The only rules which are actually adhered to by the AR platform are this:
The rifle has a lower receiver, which contains the trigger, stock, and grip. A magazine well is also included, but it might not be used, depending on how the rifle is configured. Certain variants of the AR system are designed with a sealed up magazine well, or NO magazine well (single-shot) in order to avoid legal issues with certain states. Others are designed for Glock, Uzi, or Colt 9mm SMG magazines (very similar to Uzi mags). However, the overwhelming majority of AR-15 lower receivers are set up for STANAG magazines. The standard reference example of a STANAG magazine is the venerable aluminum 30rd USGI M-16 magazine.
The rifle has an upper receiver, which contains the barrel, gas operating system, etc. In most cases, this is a gas-operated semi-automatic. In some cases, such as the 50 BMG upper receiver, the upper receiver uses a bolt.
Upper receiver choices include traditional semi-automatic uppers, single-shot bolt-action uppers, and a few exotic belt-fed uppers, in a plethora of different calibers. There are a number of them listed below; some bargains are referenced on specific pages. People are welcome to add more information for any of these calibers.
For a more general list of AR parts for sale, hit the AR-15 shopping page.