Spirex288,
Here's my quick-n'-dirty guide to trigger types:
Single Action - The trigger does only one thing: activating the previously cocked firing mechanism.
Double Action - The trigger does two things: cocking the firing mechanism and then activating it.
A single action pistol works like a rifle, where the firing mechanism is cocked by working the bolt/slide and kept locked by a safety lever until the shooter is ready to fire. Example: Colt Government Model.
A single action revolver is carried with the hammer uncocked (over an empty chamber). The shooter uses his thumb to cock the hammer (an action which also rotates the cylinder) before each shot. Example: Ruger Vaquero.
The world of double action is where things get confusing. Generally speaking, the advantage of a double action sidearm is that a shooter can fire his weapon by simply pressing the trigger without having to thumb back the hammer or disengage a safety.
Traditional double action (TDA) pistols are carried with the hammer uncocked. They also have a safety lever which can be engaged or left in the ready-to-fire position at the user's option. After the shooter uses the trigger to cock and release the trigger for the first shot, the rearward motion of the slide automatically cocks the hammer for the second and subsequent shots. When the shooter is done firing, he must use the safety lever (which also functions as a decocker) to lower the hammer back to its pre-firing condition. Example: Walther PPK.
A double-action-only (DAO) pistol has no safety or decocking levers. The pistol is carried with the hammer down and the hammer must be cocked by the trigger for each and every shot; it returns to the uncocked position automatically after being pushed back by the slide. Example: KelTec P11.
A double action (DA) revolver is carried with the hammer down on a loaded chamber. Normally, the shooter would cock and release the hammer by pressing the trigger (which also rotates the cylinder). The shooter also has the option of thumb-cocking the hammer before firing for a shorter, lighter trigger pull. Example: Ruger Redhawk.
Confused yet?
Just keep in mind that, as a rule of thumb, single action handguns are older designs with short, light trigger pulls that are generally used for competition or hunting. Double action handguns are newer designs with long, heavy trigger pulls that are generally used for police service and personal protection.
There are many variations and exceptions, so please ask if you have questions about a particular handgun.