A finisher is performed when an enemy gets injured and is briefly on the ground on one knee. The injury can be caused by shooting a non-head shot or by attacking with the axe when you obtain it after upgrading your skills to Tier 2. This also happens if you stab them in the knee with your arrow (a countering skill). Once the enemy is down, go up quickly and press the prompt which is to perform a finisher. If you want, you can easily just stack this with countering since there is an achievement for getting 25 counters. Just counter and they will be on the ground and you can finish them. You will likely get this achievement before unlocking the skills that let you counter with the various weapons.
Note: This must be done to different enemies. You cannot repeat this over and over again on the same enemy.