Having read through several thread regarding guns there is a LOT of misunderstanding and misinformation flying around. For those of you who are not gun savvy, here is a short list. This is by no means exhaustive because I’m trying to keep it basic because there are many other options in modern firearms.
Pistol actions
Single action (SA) - Only releases the hammer to strike the round in the chamber. To fire another round, the hammer must be re-cocked.

Single action automatic - The hammer must be cocked before the first round is fired. The movement of the slide cocks the hammer for each shot that follows.

Double action (DA) - Performs two functions, cocking and releasing the hammer when the trigger is pulled. Cocking the hammer manually is not required to fire the first shot. Double action revolvers are capable of being cocked between each shot if the shooter desires.

Double action only (DAO) - Similar to a DA revolver trigger mechanism but there is no single action function. For semi-automatic pistols with a regular hammer, the hammer returns to its decocked position after each shot. For striker-fired pistols, the striker is stationary through the reloading cycle.

Double action/single action (DA/SA) - The H&K USP is a good example. I owned one of these, hence the example. On many DA/SA pistols there's the option to cock the hammer before firing the first shot. This removes the heavy pull of the double-action on the first shot. Also, there is usually a de-cocker to put the pistol back to double-action.

Rifle and shotgun actions
Bolt action – Opening and closing of the breech are controlled by manually cycling the bolt.

Break action – The barrel on this type of weapon is hinged which allows the stock to rotate downward to allow loading and unloading of rounds.

Lever action – Uses a lever located around the trigger guard area (often including the trigger guard itself) to load a fresh round when the lever is moved downward and back up to locking position.

Pump action or slide action - The handgrip under the barrel can be pumped back and forth in order to eject and chamber a round.

Semi-automatic – One pull of the trigger fires one round. This is a very popular action for countless hunting rifles and shotguns. It reduces recoil and provides the convenience of not having to slide a handgrip to chamber another round.
Fully automatic – Typical firing mode for a military assault rifle. Holding down the trigger will fire the weapon until the magazine is empty. Many times the weapon has a switch to select fully automatic, a burst of rounds, or semi-automatic. Well known examples would be the AK-47, Colt M16, and the HK G36.
Machine gun – Rounds are fed into the weapon by a linked belt and the weapon can be fired fully automatic or in bursts controlled by the shooter.
Heavy machine guns fire rifle rounds and are belt-fed. Light machineguns fire smaller rifle rounds and are typically belt fed or box fed, but early light machineguns were magazine fed.
Submachine gun – Fires a pistol round in fully automatic, burst, or semiautomatic modes depending on the model. Well known examples would be the HK MP5, Thompson submachinegun, and the FN P90.
Pistol actions
Single action (SA) - Only releases the hammer to strike the round in the chamber. To fire another round, the hammer must be re-cocked.

Single action automatic - The hammer must be cocked before the first round is fired. The movement of the slide cocks the hammer for each shot that follows.
Double action (DA) - Performs two functions, cocking and releasing the hammer when the trigger is pulled. Cocking the hammer manually is not required to fire the first shot. Double action revolvers are capable of being cocked between each shot if the shooter desires.

Double action only (DAO) - Similar to a DA revolver trigger mechanism but there is no single action function. For semi-automatic pistols with a regular hammer, the hammer returns to its decocked position after each shot. For striker-fired pistols, the striker is stationary through the reloading cycle.

Double action/single action (DA/SA) - The H&K USP is a good example. I owned one of these, hence the example. On many DA/SA pistols there's the option to cock the hammer before firing the first shot. This removes the heavy pull of the double-action on the first shot. Also, there is usually a de-cocker to put the pistol back to double-action.

Rifle and shotgun actions
Bolt action – Opening and closing of the breech are controlled by manually cycling the bolt.

Break action – The barrel on this type of weapon is hinged which allows the stock to rotate downward to allow loading and unloading of rounds.

Lever action – Uses a lever located around the trigger guard area (often including the trigger guard itself) to load a fresh round when the lever is moved downward and back up to locking position.

Pump action or slide action - The handgrip under the barrel can be pumped back and forth in order to eject and chamber a round.

Semi-automatic – One pull of the trigger fires one round. This is a very popular action for countless hunting rifles and shotguns. It reduces recoil and provides the convenience of not having to slide a handgrip to chamber another round.
Fully automatic – Typical firing mode for a military assault rifle. Holding down the trigger will fire the weapon until the magazine is empty. Many times the weapon has a switch to select fully automatic, a burst of rounds, or semi-automatic. Well known examples would be the AK-47, Colt M16, and the HK G36.
Machine gun – Rounds are fed into the weapon by a linked belt and the weapon can be fired fully automatic or in bursts controlled by the shooter.
Heavy machine guns fire rifle rounds and are belt-fed. Light machineguns fire smaller rifle rounds and are typically belt fed or box fed, but early light machineguns were magazine fed.
Submachine gun – Fires a pistol round in fully automatic, burst, or semiautomatic modes depending on the model. Well known examples would be the HK MP5, Thompson submachinegun, and the FN P90.