The Dark Wing is the leader of the Ghostly Figures. They escaped from an unknown universe by use of Hope. They work for the Cows after they threatened them. He leads every ghost in the Mansion, other than Callahan.
He flies around in a circle for the first phase, and cannot be attacked physically. You must use the vacuum to bring them into a physical form and hit him three times.
For the second phase, he can be hit already, so he must be dealt enough damage in a short enough time before he regenerates health.
Not much is known about the Dark Wing's personality other than that he likes to joke.