"Double action" means that you can fire either from pulling the trigger with the hammer down, thus cocking it by pulling the trigger, or by firing it with the hammer already cocked. Sigs are a good example of this type of action.

"Double action only" means that the hammer follows the slide down every time and every time you pull the trigger you cock the hammer. Just like pulling the trigger on a revolver with the hammer down.

"Single action" means that the hammer must be cocked, either manually or by the slide cocking it, before it will fire. The Colt 1911 is a single action auto.

Then you have guns like a Glock or Springfield XD that are striker fired and essentially function just like a single action auto in that the slide moving rearward cocks the striker inside the gun. However, you can not manually cock the hammer since there isn't one but you can short stroke the slide to recock the striker. Even though they function more like a single action I think they are classified as double action only.