I also don't fully understand hammers on semi-autos. (Why would you need or want one? How does it function on a semi?)
My God, you are young. Oh, that's right, I'm old.
Single action autos are autos that requrire the shooter to cock the hammer or striker at least once. The recoil from the handgun cocks the handgun there after. An example in a hammer gun is the 1911, BHP, CZ 75B (single action model). An example in a striker gun is the P7. It is cocked by the squeeze cocker once, the P7 cocks the striker there afterwards. There is no way to simply pull the trigger on an uncocked single action handgun, and have it cock the hammer or striker.
A Double Action/Single Action commonly written as DA/SA is generally a hammer gun. This type of handgun can be fired or operated by cocking the hammer or simply pulling the trigger. There are many examples which range from CZ75B (DA/SA), HK USP, etc. Once the shooter fires the handgun by either cocking the hammer first or pulling the trigger, the hammer then is cocked by the recoil of the slide.
Now for the more recently developed striker handguns, termed
safe-actions, the Glock comes to mind for most. There is no manual safety on a Glock, but rather a series of safety devices, which disable the gun's ability to fire unless the trigger is pulled. This is neither a doulble action, nor a single action handgun. The striker is partially under tension. The shooter, by pulling the trigger, completes the remaining tension needed while releasing the striker. You can look at this as double action light. Each sequential shot is the same.
Another catagory is the double action only handgun (DAO). This handgun has one and only one mode of firing. The shooter must pull the trigger, which cocks the hammer, each time he/she fires. This differs from a single action only in that the hammer must be cocked first, before the trigger is pulled, or the DA/SA handgun which can be fired SA or by pulling the trigger with the hammer cocked or by pulling the trigger with an uncocked hammer.
So why so many different varieties. Simple, its how we keep you young shooters in check. We know more than you.
Just kidding.
The SA only handguns came from an era when liability and litigation were not the issues they are now. In fairness, SA only firearms were developed before much of the training techniques we have now.* With SA only, the shooter gets the same pull each time he/she fires. The pull is generally much lighter, and with less pull distance over a DA pull. Those that shoot DA/SA autos, generally do so, because it becomes less likely that that first round will be discharged due to sypathetic muscle contraction. In other words, it takes much more pressusre to pull off that first round, therefore it is unlikely you will discharge your handgun from being startled or surprised. The DA only handuns take it to another level. Every pull is long and requires about 12-15 lbs. of pressure. Some like this, because every shot is consistant. The safe-actions such as Glock are some where in between. Those handguns operate like a SA, but require a longer pull than most SA autos; the idea is to keep an unintentional shot from being fired.
Hope this wasn't too confusing.
* This can be as safe as other handguns, if the shooter is trained properly, and is willing to keep up the practice.