You will arrive on the Pirate Ship after leaving Outset Island. You can walk around and talk to whoever you please, but to progress in the game, head through the door opposite the steering wheel.
Niko's Minigame
Walk down the stairs and you'll meet Niko. He will show you a minigame in which you must raise platforms and then swing from ropes to get to the end. When he is finished explaining, hop to the first platform. Hop at the rope, and swing to the next platform, and the next, taking your time, and so on until you reach the end. If you fall off, Niko will explain how to turn when you're holding a rope, by holding 
. The advice is to line up to the rope and at least swing twice before you hit the platform. You will overstep the platform if you swing once and jump the platform. The last part if the platform did lower itself, line yourself where Niko is and take a few swings onto the final spot to finish this puzzle.
The Legend of Zelda: The Wind Waker/NewItem
When you astonish Niko by getting to the end, he will give you the contents of the treasure chest behind him: the Spoils Bag. This item allows you to keep various items that you get lots of. It becomes useful later in the game. After you get it, Tetra will call you up to the top of the ship, so go up the stairs you came in on and out the door.
Once outside, climb up the ladder in the middle of the ship to the top, and you will begin a cutscene where you are launched to the Forsaken Fortress.