Refer to "Arrr!" for how to board a vessel.
This needs to be done while manually fighting a naval battle and will require a decent sized convoy with maximum sailors and weapons. When it comes to boarding the enemy ship, try and wait until the enemy has between 25%-50% the number of sailors you do before boarding. This will ensure you don’t waste sailors but still have enough enemy sailors to fight through. Each time you board you will use up your handheld weapons and lose some sailors, so you will need to resupply your convoy with weapons and sailors in town as you get low. This will require some grinding but will carry between saves.
TIP: A good strategy is to ensure you have plenty of pirate hideouts scattered around the map. Simply sail around attacking the pirate convoys but not the hideouts, as the pirate convoys will respawn.