SA = An M1911, Browning Hi Power, Tokarev, etc. The hammer must be retracted to the cocked position for the first shot. This will happen automatically when the slide is retracted to chamber the first round. The hammer can however be carefully lowered, after which it must be cocked to fire the firearm. This is not recommended on a typical handgun such as a Hi Power or M1911. An original Tokarev has no manual safety and would typically be carried with the hammer down on a loaded chamber.
SA/DA = A S&W Model 39, SIG P220, Walther P-38, etc. Once the chamber is loaded by retracting the slide, the hammer can be lowered safely using a decocking/safety lever of some sort. When the gun needs to be fired, pulling the trigger will retract the hammer and allow it to fall, causing a round to be fired and the hammer again cocked by the action of the slide. If desired instead, the hammer can be manually cocked with the thumb. There are a few DA/SA handguns, such as the CZ75 which can also be carried cocked and locked (hammer cocked, safety on) like an M1911.
DAO = The hammer can be cocked ONLY by action of the trigger. When the slide goes forward after ejecting an empty case and chambering a new round, the hammer returns to the lowered position. The trigger action is exactly the same from shot to shot, retracting and allowing to fall, the hammer. A hammerless striker fired pistol can have the same sort of trigger mechanism which retracts and releases the firing pin itself with each pull of the trigger.