Pirate hideouts are the home bases of the pirates who operate in the Caribbean. There are no pirates initially, though throughout the game, new pirates will begin to operate. These pirates will raid ships that pass by their patrolling areas, so it is best to clear our these hideouts to allow your trade convoys to operate without interference. Destroying a pirate hideout also results in a cash reward, however this may not be an effective means of earning money if your ships sustain heavy damage or are sunk during the initial phases of the attack.
Pirate hideouts are initially hidden, effectively allowing the pirate to gain a foothold before the player has the chance to destroy the hideout. Although pirate hideouts appear in standard locations throughout the map, a convoy (doesn't matter who it belongs to) must observe the pirates either entering or leaving the hideout before the location is marked on the map.
Once marked upon the map, a pirate hideout can be attacked by any convoy that contains escort ships. The attack has three stages.
The first stage is a sea battle versus any ships in the harbour, which you must win to advance to the harbour. If the pirate's ships has no ships, or they are out when the attack occurs, there is no sea battle. Pirates often sail uncommon ships that cannot normally be purchased, and these can be boarded as normal. Sinking the ships will lead to a cash reward when you finish clearing out the hideout.
The second stage is a harbour battle against the harbour defence cannons. You cannot attack by land as you could when attacking a town. The number of defence cannons will increase the longer the pirate is allowed to operate, which is incentive to take down pirates as early as possible (nine or ten cannons is a nightmare that requires skillful tactics and/or sacrificing many ships to overcome).
The final stage is a sword duel versus the pirate himself. Winning this final battle ends the pirate's career and nets you a sizeable cash reward. In addition, any governor will give you your reward for any sunk ships.
Be aware that any pirate ships that were out during the attack (or escaped during the sea battle phase) are liable to re-establish under a new commanding pirate in a new location soon after the destruction of the hideout - it is best to eliminate all ships prior to destroying the hideout to prevent this. (They'll re-establish anyway, but they'll take a lot longer if no ships escape the hideout.)