It's not hard to understand once you get the hang of it
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On a traditional hammer-fired semi-auto, there is a spring in front of the firing pin pushing it to the back of the slit. Keeping it pushed back ensures that when it's in it's rested state, it's not sitting on the primer.
The hammer is used to hit the pin with enough force to throw it forward (compressing the return spring) so that the pin impacts the primer and detonates it.
The hammer itself - when cocked, is held back by a sear. When you press the trigger the sear is moved off the hammer and it flies foward.
Now, hammer fired designs come in 3 main flavors:
Single-action - the hammer has to be cocked prior to firing the gun. When the first round goes off, it cycles the action, loads the new round, and leaves the hammer cocked again for subsequent shots.\
Double-action - the hammer can be cocked as with a single action, but you also can fire starting with the hammer down. This will be a longer heavier trigger pull itself will pull back the hammer as well, and when the trigger comes all the way rearward the hammer is released and fires. When the gun cycles though, it leaves the action pre-cocked just like a single action. Subsequent shots are like a single-action and have the lighter trigger pull.
Double-action only (DAO) - With this type of hammer fired gun, you cannot pre-cock the hammer (there usually isn't even a grip on it to do so). You have to perform the long heavy double-action trigger pull like described above, but after that round fires, the hammer comes back to it's original rest state. The result is that you have to perform the long heavy DA trigger pull for every shot.
Striker fired guns are a bit different. Instead of a spring pushing the firing pin back, it naturally pushes the pin forward instead. HOWEVER, when the slide is racked, the sear actually engages the firing pin (called the striker in such designs) and the sear holds the pin off of the chambered round. When you pull the trigger, the sear is released and the firing pin slams forward under it's own spring pressure to detonate the round.
Basically anyways - depending on the design the pin may need to be pulled back by the trigger pull (similar to a double action hammer) some distance before it is dropped. Glock "half cocks" their striker for example - meaning it sill has to be pulled back about half-way before it drops. The Springfield XD fully-cocks their striker on the other hand - meaning that it merely drops. Other designs may only leave the striker cocked enough to keep it off the primer (the S&W Sigma does this IIRC).
When you hear "double strike capability", it basically means you can pull the trigger again on a round you snapped on and it'll snap again. All DA and DAO hammer guns will do this since they fire from a rested state anyways. SA guns won't. MOST striker fired guns won't, because they require the slide to leave them in a partially cocked position. If the striker fully falls most require the slide to rack to be ready again. There are some exceptions though (the Taurus 24/7 G2 is a striker fired gun with double strike for example).
There are also a few odd mixes. The Ruger .22 handguns for example. They utilize a bolt with an internal hidden hammer. Technically it's like a single action, but since the hammer is hidden, the gun is always pre-cocked after racking the bolt. If you snap on a round, you have to cycle the bolt again as there's no way to manually recock the hammer.
All just different ways to do the same thing. Personally though, as someone who owns a gun with almost all the designs described above (except for DAO hammer), I am a big fan of the regular DA hammer guns. They have that safe, long, heavy trigger pull for your first shot if you want it, and you still get easy, light followup trigger pulls at the target range. To me SA is outdated, and DAO and striker guns are moreso for guns that you don't intend on firing often (because their systems are moreso geared towards safety, not comfort or accuracy).
