Call of Duty: Modern Warfare has introduced a fun new mechanic. You can now get an execution that kills enemies in PvP. If you want to punish an annoying camper, you have a new weapon in your arsenal. This is a one-shot kill with an added animation, so there is risk involved if you want to experience the thrill of the reward.
Call of Duty: Modern Warfare - How to Perform Execution Finishing Moves
To perform a finishing move, you must first get behind your target. It's often easy to get left behind because campers have a habit of getting tricked into looking at their sites. They also fail to pay attention to audio cues because they spend too much time predicting the next sweat.
So, sneak up behind someone, and then you have to hold down the melee attack button. Pressing it will just slide your knife or gun out. If you hold down the melee attack button, you will start the last move.
Because you need to stand by for a short time, the enemy might become aware of your position and attack you. If this happens you will need to spam your melee attack to take them out before they can kill you. When the animation starts, it's done and your opponent will meet a terrible end.
You can also use some sneaky tactics, like hitting an enemy with a flash bang, stepping in behind them, and making the last move. The result is the same, but your opposition has no choice but to admit they have been outdone.