Single action - the trigger serves a single purpose, releasing the manually cocked hammer.
Double action - the trigger serves two purposes, cocking the hammer to fire and releasing a manually cocked hammer.
These definitions were all well and good up to the middle of the 20th century. Things started to get weird when DAO (double action only) handguns made the scene. Double action only? WTH? Double action was by definition TWO different available functions. Now a trigger that only does one thing is DAO? You never see anything referred to as SAO, do you? Striker fired guns REALLY clouded the issue. As their is no hammer, what do you call them? Some striker fired guns preload the striker so the first and subsequent shots are really single action. Some do not load the striker so the first shot is more double action like. Some are kind of in between and even worse, some have options ranging from loaded to unloaded strikers (P-99).