Normally only found in pistols, double action refers to the operation of the trigger - it both cocks the hammer and releases it. The subsequent discharge of the weapon may - or may not - then chamber a new round. Double action triggers require a much heavier pull than a single action trigger, leading to less accurate shooting, but have the advantage of allowing the user to bring the weapon into play much more quickly. Note that the cocking stroke of a double action trigger only cocks the hammer rather than cycling the whole action (as happens when a weapon is cocked by cycling the bolt), so if it is fitted to a self loading weapon (or even a revolver) it will not be able to chamber a round. This does, however, allow the weapon to be carried more safely with a round in the chamber.
Double action only weapons are not particularly common - except in the case of revolvers, where a fair percentage of designs work like this. As noted above, this trigger action is all but unheard of in long arms.
For the best compromise between single and double action, see dual action.