Single-action and double-action refer to how a gun's mechanism operates when the trigger is pulled. The "double" in double-action means the trigger performs two functions: cocking, and then firing the gun. The hammer cannot be manually cocked back; only the pull of the trigger can cause that to happen. Most weapons are single-action (SA) but some handguns (pistols and revolvers) can be double-action (DAO aka double-action only) or single-action/double-action (SA/DA).
A mechanism is the trigger, hammer and safeties considered as a single unit. The action of a weapon defines how the mechanism is built and used. The types of action (single/double) vary based on the function performed by the trigger.
In a single-action, when the trigger is pulled, the hammer is released, causing the cartridge in the chamber to fire. They are called “single action” because only one mechanism – the release of the hammer – occurs when the trigger is pulled in order for the gun to fire. Single action (SA) revolvers must be cocked manually prior to each shot, Single action automatics such as the 1911 patter pistols must be cocked manually prior to the first shot, but on subsequent shots, the hammer is cocked automatically by the reward travel of the recoiling slide.
Double-action handguns come in various forms. As the trigger is pulled on a double action only (DAO) and striker fired handguns, the hammer (or striker) is drawn back(and, in revolvers, the cylinder is rotated). Once the trigger has been pulled back to its "break" point, the hammer is released and dropped on the cartridge, causing it to fire. All shots are fired with the hammer initially uncocked as the hammer has no SA lock point.
In double-action/single action (DA/SA) handguns, or “traditional double action” guns, the trigger is identical to a DAO revolver, but the recoiling slide automatically cocks the hammer after the gun is fired. For each subsequent shot, the trigger functions as a single action. Both DA/SA pistols and DA revolvers with exposed hammers can be fired SA whenever the shooter so desires.
edit Trigger pull
A single action revolver has a light and smooth trigger pull, as it only needs to drop the hammer. This allows for more accurate shooting.
A double action revolver has a heavier, longer trigger pull, which can be detrimental to accuracy.
A video showing how to shoot using a single and double action gun:
Reloading is pretty much identical for automatic pistols regardless of action type. Drop empty magazine, insert full magazine, rack or drop slide.
A single action revolver usually requires the user to open the gate at the back of the cylinder to insert each round and then rotate the cylinder to bring the next chamber into use.
A double action revolver can be reloaded more quickly, by swinging the cylinder out of the gun frame, and then using a speed loader or hand-loading the rounds.
Some consider double action weapons to be safer than single action weapons, but modern innovations have made it very difficult to make a well maintained firearm of either type unintentionally discharge. Safety Notches, Firing pin blocks, hammer blocks, transfer bars, trigger disconnects, grip and trigger safeties are all good examples of these.