Double-action simply means that the trigger completes two functions; cocking the firearm, and releasing the hammer to fire the firearm. This is opposed to single-action where the trigger merely releases the hammer, which must be cocked by some other mechanism--either the movement of the slide in an autoloading handgun, or via the operator manually cocking the hammer in a single-action revolver. Because the double-action mechanism must cock and release the hammer, the trigger pull is usually longer and heavier than the typical single action trigger pull. Some lawyer driven mentalities hold this as safer than the lighter trigger pull of the single action firearms.
Among double-actions, their are two disctinct types. Most are so-called DA/SA or "traditional DA." This includes pretty much every double-action revolver without a bobbed or enclosed hammer because they can be fired DA, or the hammer can be manually cocked to fire the handgun SA. In autos, a DA/SA will typically have a long, heavy DA trigger pull for the first shot. The slide will cock the hammer for subsequent shots, allowing for subsequent shots to be fired with a shorter, lighter SA pull. These pistols usually have a de-cocker that allows the hammer to be safely dropped before the pistol is returned to its holster.
Double-Action Only (DAO) means the pistol can not be cocked and fired in single action. Every pull of the trigger will necessitate that the hammer be cocked and released. Any revolver with a bobbed or enclosed hammer would be considered DAO.
Striker vs hammer fired is a completely different argument. Striker fired mostly means that the trigger mechanism acts on a striker, which is basically just a modified firing pin in most cases, rather than a hammer, then a firing pin. There are striker fired rifles, like the CZ VZ-58 and most bolt actions, and striker fired handguns, most noteably, the Glock. There are single-action striker fired handguns, like the Kimber Solo. Most, like the aforementioned Glock, qualify as neither single, nor double actions.
In the Glock, the movement of the slide cocks the striker assembly to a little over what would normally be referred to as half-cocked. This is done so that even if the striker safety and other passive safeties fail and the striker is released when the pistol is dropped, for example, the striker still theatrically doesn't possess enough energy to fire the cartridge, thereby making it mechanically impossible to cause the Glock to fire by dropping or jarring it. The striker is cocked the rest of the way and released by the movement of the trigger, much like a double-action. However, in order to qualify as a double-action, you must be able to fully cock and release the hammer or striker through the movement of the trigger. All true double-actions will have so-called "double strike capability" because all that is required to cock and fire the pistol is for you to pull the trigger. This is not possible with the Glock, which requires movement of the slide to bring it to its stored "half cock" position. Therefore, the Glock is technically neither double-action or single-action. The ATF may group the Glock in as a double-action but this is because a) no one ever accused anyone in the esteemed federal agency that tried to ban a 14 inch shoelace as a machine gun as being smart and b) because more than likely, there was no "other" option to mark on the paperwork, just single-action or double-action, as prior to the Glock, these categories likely sufficed. Still, if you're holding a pear, but the only options on the sheet are "apple" and "orange," you can mark "orange" if you want, but it doesn't make your pear an orange. And the Glock is absolutely not a double-action, period.
