Basically, Single Action means that you can only fire when the hammer is clocked. If the hammer is not clocked, either manually or by other means (will explain in a moment), then pulling the trigger will not clock the hammer, and thus, the gun will not fire.
Double action operates like Single Action in that the gun will fire when the hammer is clocked, but it will also clock the trigger, and then release it to fire when it is not already clocked.
Now, here's the catch. Most semi-auto pistols are single action. What this means is that as the bullet is being fired, the energy from the bullet will push the slide back. This ejects the cartridge, chambers another cartridge AND clocks the hammer making it ready to fire the next round.
Single Actions are more accurate due to the light trigger pull weight leads to less strain pulling the trigger. Firing through double action is less accurate, because you are pulling more weight and trying to keep the gun steady.