When in a Duel, War, Empire Battle (EvE), or Boss fight and your opponent is below some level of Health, you will be given the chance to execute them. You will initially have access to three execution methods, and each time you use one, the odds of success for that method may increase. Increasing the odds of success of some methods to specific levels unlocks other methods.
The level of health at which you can attempt to execute your opponent is increased if you have Scorpion. For empire battles, it is also increased if you have the Decapitate assassin skill. For Bosses, it is reduced if you have Anson.
Each execution method has a percentage chance for success, which is how likely you are to kill the target each time you use that method, and a separate percentage chance to earn some sort of bonus. The first percentage increases with practice, while the second is fixed for each method.
There are Achievements that can be earned for certain numbers of successful executions, ranging from 5 to 10,000.