The M1 cabine fires a .30 carbine round similar in power to a .357 magnum pistol bullet. It was intended, to a certain extent, to replace/supplement pistols so that troops that are traditionally armed with pistols(officers, MPs etc) would have a better chance at hitting foes and still be equipped with a relatively lightweight firearm. The M1 carbine also overlaps the submachinegun (i.e. the Thompson) to a certain extent also. There were full auto versions of the M1 carbine, although I am not sure of their reliability. The M1 carbine seemed to be a pretty good idea, but suffered from often needing multiple hits to incapcitate and enemy.
An M1A is a full size rifle that fires 7.62 x 51 (basically .308) and is similar to the 30-06 M1 Garand. The M1A is the civilian semi automatic version of the military M14. It takes a 20 round box magazine. A Garand takes a 8 round clip that is inserted inside the rifle. These are very different rifles from an M1 carbine. They are heavier, much more accurate and fire a much more powerful bullet. They are designed as a main battle rifle for infantryman v. the M1 Carbine which is basically intended to serve a somewhat similar role to a pistol.