A short way into the game, you will notice that a B prompt will appear over enemies. This is your cue to get in close and execute them. Just press B and the execution sequence will happen, triggering the achievement.
Note that on later levels this allows you to ride the scorpions and large enemies. X and Y still carry out attacks, and B can be pressed to finish off a ridden enemy.
And at one point you will be given the ability to convert enemies. Again, get close to a target with the B prompt over its head, press B and then bash B to possess the target and convert it to your side. It will then fight nearby enemies, or can be finished off for bonus mana/strength.