I have so many hours in Skyrim already.. yet I still don't understand how the "Instant Kills" work... When I say "Instant Kills" I mean like cut-scenes (shoving my sword through their chest, or cutting their head off, etc etc).
So how does it work? I get so many opportunities to do it.. yet instead I just continue to slap the enemies with my sword.. And I know it has nothing to do with their health.

It has to be a blow that would kill the person and its random.

I believe it's just a normal killing blow, but there's a random chance it will perform one of the animations instead of a slash.

It's extremely common for Assassins due to the massive sneak attack bonuses they can get. My Assassin will get an animation on 9/10 guys he assassinates.

Depends on a couple factors:
1. Positioning
2. Weapon used

For the throat slitting one, I find that it's best if I use one dagger, power attack (regular works as well if it's enough damage), directly behind them, they have to be fully upright (can't be sitting, can't be falmer (who never stand fully upright)), and I have to be hidden.

As for getting the kill animations with a sword, I'm not sure since I haven't use one that much. But hopefully my advice involving a dagger will help regardless.

That's a stupid design.. maybe instant kills "Random."
I sneak up on someone, with two daggers drawn, taking them by surprise, get right behind them, hoping to cut their throat.. but instead I just slap them over the head with my blades..
If I'm going to kill someone in one hit anyways, at least make it look cool.

I really enjoy the axe ones. Mostly because they're very short and to the point.

I noticed if you wield with your weapon in your left hand, you don't get the killing animations. So if you're doing that... that could be your reason.