To board a vessel you need to play a naval battle manually. Before engaging the ‘Board Ship’ option it is a good idea to wear down the enemy ship’s defences by using chain-shots to destroy the masts and slow the ship, and scatter shots to decimate the enemy crew. As is recommended several times throughout the game, equip your sailors with a 4:1 ratio of cutlasses and muskets.
Once you wear the defences down select the board option in the ammo menu by pressing then try and bring your ship inside the blue circle that appears. Once you are inside the circle a board reticule will appear and begin to fill up. During this you need to keep your ship inside the blue circle or the reticule will restart. Once the reticule is green, the boarding fight will begin automatically. Provided you took out the majority of the enemy crew, your crew will win the fight and the achievement will unlock at the end of the battle, provided you win the overall battle.