The Hindenburg III was an airship that made its fictional maiden voyage to New York on 7th October 1939, an event which features in the opening scene of Sky Captain and the World of Tomorrow. It is unknown why the ship is the third Hindenburg airship, as in reality there was only ever one.
The Real Hindenburg
LZ 129 Hindenburg, to give it its full title, was an identical, real life, airship which made trips between Europe and America many times in the 1930's. This airship famously caught fire over New Jersey as it prepared to dock at Lakehurst Airfield in the greatest zeppelin disaster of the 20th century. This tragic event spelled the end for the airship and the golden age of the zeppelin was over.
In Sky Captain
In the film, the airship arrives over New York carrying the fleeing Dr Jorge Vargas, who disappears just as the airship docks with the Empire State Building. The airship of the film lacks the tail swastikas present on its real-world counterpart, showing the changed nature of events in the alternate 1939.
Inside, the Hindenburg featured many tasteful design features and aesthetic comforts. The observation deck, from which Dr Vargas dissapears has a row of windows which run along a quarter of the airship's 803ft length on both sides.
After Dr Vargas' dissapearance, the airship does not features again in the film and it can be assumed that it continued to serve as a passenger airship well after the events of the film.