Double-action can be fired one of two ways: either by cocking the hammer and pulling the trigger, or by pulling the trigger all the way through the action. Single actions must be cocked first.
The famed Colt Peacemaker (Single Action Army) is an example of a single-action pistol. Most modern revolvers are double-action.
And then you have DA/SA firearms, semi autos, where the first shot is shot as a DA (double action), and then after the first shot the slide automatically cocks the hammer by its movement, and so the rest of the rounds can be shot SA. Unless you decock the hammer. Then the DA/SA sequence starts over again.