Standard is the same as most other CoD titles. There's an invisible bubble around the player where your aim will slow down a little bit when you pass over it. On standard, this extends maybe one player width either side of the enemy player and the slowdown amount is quite small. Precision the "bubble is much much smaller, slightly narrower than the player model itself, but the slowdown is more severe. Focussing is somewhere between the two. What none of these assist options do is "snap" you onto target. If you ADS while your crosshair is off target, you will still be off target when you ADS.
Standard and Focussing also give a small amount of "rotation", which is why strafing is so common on console. Precision apparently doesn't but that might have changed in recent patches.
Personally I use Standard with Dynamic response curve, works best for me. The slowdown with precision is way too jarring for me and I'm less accurate with it.
