Watcher Battleships are covered with Watcher Stationary Turrets (some of which are hidden under the hull of the ship to protect them), armed with a powerful main cannon, carry a contingency of several Watcher Flying Discs, and are able to launch Watcher Mines (which somehow also double as turrets until fired at you). They are the pride of the Watcher fleet.
The only time you encounter one in the game is the end of the first chapter of the third Episode, in which a Battleship emerges from the water in the Broken Path and assaults The Ark, and Will Grey must find a way to destroy it.
To take out the Battleship, focus on the turrets covering its exterior first; also, take care of any Discs it may launch during this period of time. Make sure you also take care of the Watcher Mine Turrets embedded in the hull of the vessel, which not only fire shots at you, but can also be launched in your general direction as an actual mine and home in on you if you are too close, making them a very serious threat if not destroyed.
Destroying the "mine deployment turrets" also lowers the shields protecting the power cores of the Battleship (since the "mine deployment turrets" were apparently also powering the shields). Now you can finally exploit the Battleship's weak points (the power cores), located at the two ends of the craft. By rewiring two control panels at both ends of the vessel, you can cause the power cores to come out from the hull of the ship; shooting these power cores cuts off the power to the entire ship, causing it to free fall back into the water to which it emerged, and ends the mission.