In a semi-automatic, there are three basic action types: Single-Action, Double/Single-Action, and Double-Action-Only.
With a Single-Action semi-auto, the hammer/striker is cocked either manually or by the slide riding back over it when the gun cycles. The advantage of a Single-Action is that you get a relatively light, short trigger pull that is consistent for every shot.
A Double/Single-Action semi-auto is one that can have its hammer/striker cocked in the same way as a single-action, but a longer, heavier trigger pull can also be used to fire the gun when the hammer/striker is not already cocked. The advantages of a Double/Single-Action are that many people feel more comfortable carrying a gun of this type without a manual safety and that this action type allows for a second strike in the case of a misfire without having to manually cock the hammer/striker or manually cycle the gun.
A Double-Action-Only is pretty much exactly what it says. The hammer/striker cannot be manually cocked and is not cocked by cycling the slide. As such, the shooter will have the same longer, heavier trigger pull for every shot. The advantages to this system are the same as those of the Double/Single-Action but you also have a consistent trigger pull for every shot like a Single-Action (though you give up the short, light trigger of the Single-Action). Also, some people feel that this setup is the safest because the shooter must pull through the longer, heavier, more deliberate trigger for each shot.
There are also guns like Glock which don't fit concretely into any of the three categories. Glock's "safe-action" is considered DAO by the ATF, but in actual practice it is much more like a single-action. A Glock's striker is partially cocked when the slide is cycled and brought fully back and then released by pulling the trigger. A Glock does not have a second-strike capability and, unless modified, also has a relatively short and light trigger much more like that of a single-action pistol.
In a revolver, you also have Single-Action, Double/Single-Action, and Double-Action-Only, but they mean slightly different things.
A Single-Action revolver can only have its hammer manually cocked by the shooter. This means that a SA revolver must be manually cocked for each and every shot. The advantage to this system is that, due to the designs of SA revolvers, the gun can be made smaller, lighter, and simpler than a DA revolver without sacrificing strength. The mechanism of a SA revolver also allows the frame to be made in such a way as to make the gun balance and point better in some people's hands.
A Double/Single-Action revolver can be manually cocked just like a SA revolver, but it can also be fired by simply pulling the trigger although the pull will be longer and heavier. A DA/SA offers the shooter faster repeat shots since manually cocking the hammer is unneccessary, but the shooter also retains the option of a shorter, lighter SA shot if so desired.
Just like with a semi-auto, a Double-Action-Only revolver is pretty self-explanatory. The hammer cannot be manually cocked and the shooter must pull through the longer, heavier DA trigger for each shot. The advantages to this system is that many people feel it to be safer because the more deliberate trigger pull is required for every shot and that it forces the user to practice DA shooting exclusively (DA shooting is the type considered by most to be best for self-defense). A DAO revolver can also have its hammer spur removed or have the entire hammer enclosed within the frame since there is no reason to manually manipulate the hammer. A bobbed or fully enclosed hammer makes the gun less apt to snag on clothing or holsters when the user attempts to draw it quickly.