Single Action OnlyAbbreviated SAO
Your two most common types if SAO guns are 1911 pattern semi-automatics and your "Western" revolvers" such as the Colt Single Action Army. A lot of people get confused on this as they mistake action type and trigger type. With both the 1911 and single action revolver, the hammer must be cocked in order for the gun to fire. Pulling the trigger causes one action, that of the hammer to fall and make contact with the firing pin. To confuse you even further, there are some single action only guns that are striker fired as they are fully cocked by the manual cycling of the slide or other operation. Two examples of single action, striker fired guns are the HKP7 family and the Springfield XD (includes XDM) line. There are some that will disagree and say that striker fired guns are not SAO. But looking at the requirement that pulling the trigger facilitates only one action, the release of "X" that initiates the firing sequence, YES they are single action and are even considered SA by the IDPA.
Double Action Abbreviated DA.
Most modern revolvers are referred to as double action, despite the fact that they are still capable of being fired in the single action mode, why this is I don't know. Double Action indicates that pulling the trigger caused two actions, 1: the cocking and 2: release of the hammer. One action=two results. Common double action revolvers are the Ruger GP100 and Redhawk
Double Action Only, abbreviated DAO.
DAO guns can not be manually cocked, they are cocked and released by trigger manipulation. DAO guns can be revolvers or semi-automatics. Glocks are erroneously referred to as DAO as the trigger must be pulled to fire the gun, THIS IS FALSE. Glocks and many other makes/models are partially cocked, striker fired guns, more on them later.
A DAO gun has the following traits.
1: Is only cocked by trigger manipulation and the trigger pull is the same for every shot.
2: In semi-auto guns, the hammer will go back to rest (hammer down behind the slide) after each shot. Remember, the trigger cocks and releases the hammer, not the slide in this case.
3: As the trigger is the driving force to cock and release the hammer, second strike capability is there. If you have questions on second strike capability...try Google, I'm not covering that in this piece. Common DAO guns are the S&W 642 and KelTec P3AT
There are some different types of DAO, such as DAK and LDA from Sig and Para Ordnance respectively, I will cover those in brief at the end.
Traditional Double Action / Single Action Abbreviated DA/SA.
This type of trigger is what is found on the Beretta M9,and other semi-automatic pistols. The first shot can be double action or single action IE one pull of the trigger will cock and release the hammer. If the hammer is manually cocked, the pulling of the trigger will release the hammer. As the M9 is a semi-automatic pistol, the slide will re-cock the hammer after the gun is fired. With a DA/SA type of gun, all subsequent shots will be fired single action until shooting stops and the hammer is de-cocked.
Partially Cocked Striker Fired / Striker Fired
Glock, Kahr, some S&W models and others operate on this type of mechanism. What sets these aside from DAO guns is that the cycling of the slide partially cocks the striker and the pulling of the trigger finishes cocking the mechanism and releases the striker to make the gun go bang. Remember what I said before, some striker fired guns fall into different categories.
Now that we go the basics out of the way, we can take a look at some different "sub-types" of triggers. As mentioned previously, out there in the world there are these two strange beasts that are called DAK (Double Action Kellerman) and the LDA (Light Double Action) which are found on so equipped Sigs and Para LDA models.
SIG released an altered version of the double-action only (DAO) pistols called the DAK (for Double Action Kellerman, after the designer of the system). The DAK capability is available in 220, 226, 229 and 239 models. When firing the pistol the first trigger pull is 6.5 lbf (compared to 10 pounds for the standard DAO). After the pistol fires and the trigger is released forward, the trigger has an intermediate reset point that is approximately halfway to the trigger at rest position. The trigger pull from this intermediate reset point is 8.5 lbf (38 N). If the trigger is released all the way forward, this will engage the primary trigger reset and have a trigger pull of 6.5 lbf (29 N). To engage the intermediate reset, the trigger must be held to the rear while the slide is cycled, either manually or by the recoil of a round being fired. The United States Coast Guard has adopted this firearm as its PDW (Personal Defense Weapon), replacing the older M9 pistol.
As there are two different pull weights?? this is not a conventional DAO.
Without going into gross detail, the LDA is a super smooth, lightweight trigger. Generally speaking most DAO guns have a heavy pull and no safeties. The Para LDA incorporates the 1911 pattern thumb safety thus allowing for a very light pull but still has the hammer going back to rest after every shot, but if I recall correctly, the LDA has no second strike capability and needs the slide to cycle in order for the trigger to do it's thing in cocking and releasing the hammer.
HK refers to their DAO as LEM (Law Enforcement Modification) which is thought to be another form of lighter double action, but it is really another pre-cocked hammer system, you can read more about it at HK's website
It's a lot of info to take in. Each method has it's advantages and disadvantages. If you'd like to read more on the types of trigger actions on semi-autos, I would definitely recommend checking out Automatics: What Action Type for Me? by Stephen Camp.