This island is home to the Legendary Pokémon Darkrai. Currently only ways to get this Pokémon are to walk through walls with an Action Replay cheat or start the event with receiving the 'Member Card' from the man wearing green in the Poké Mart. To do so, walk to the right from Fullmoon Island and eventually you will reach the Island.
After receiving Member's Card from a Nintendo event, go to Canalave town and go to the inn in the north of the town. once you enter, a weird man will put you to sleep, and you'll get to Newmoon Island.
Go through the entrance to the forest clearing. You will see Darkrai hovering there. Walk up to him and challenge him.