DA is double action, the trigger cocks and fires.
SA is single action, you have to cock it manually, the triggers releases the hammer ( like the Colt Peacemaker of cowboy movies where you pull the hammer for each shot)
DA/SA is double/single. If you rack the slide it cocks the hammer. If you pull trigger it is SA. If you decock it, lower the hammer, the gun can fire as a DA. The trigger pull recocks and fires it.
DAO is double action only. Each trigger pull cocks and fires.
Single actions have a lighter pull since they just release the hammer. The DA is a longer pull, and usually harder. The DAO is sometimes considered a safety feature, a long hard pull, no hair trigger.
My Ruger P89 has a decocker and is DA/SA. My Ruger LCP is DAO, I think. It sure feels like it. The long trigger makes it so it wont fire until you are good and ready.