Man, this comes up so much it should be a FAQ. Here's the deal:
When Colt bought the AR-15 rights, they developed a family of weapons to sell based on them. They called it the CAR-15 family. These inlcuded the rifle, carbine, SMG, pilot's survival, squad support, etc. They were all CAR-15s, which stood for Colt's Automatic Rifle. It's a marketing ploy to use the old "AR" (which was Armalite Rifle) brand with the Colt name.
The Army bought and tested the Sub-machine gun version of the CAR-15 as the XM-177 series of guns in various versions. The CAR-15 trade name stuck with that weapon, though it was called various things by troops, like "shorty", "CAR-15", "Commando", etc.
When various aftermarket companies came out with cvillian versions of the shorter version of the standard rifle (i.e. the CAR-15), it was referred to as the CAR-15 generically, the same way the standard rifle is an "AR". Remember, back then there was a larger, younger, Vietnam veteran population in the gun buying public.
Oddly enough, Colt dropped the CAR designation for the family of weaopns and remaned them the M16 family, to build off the now famous "M16" moniker. They included all sorts of different versions as well. Some have sold (like the M4, and the 9mm SMG) and others have not.
So the real answer is that historically, "CAR" stands for Colt Automatic Rifle. CAR-15 also is a generic term for a carbine version of the AR/M16.