"Perhaps America's most lyrical monument to the dawn of the jet age", the TWA Flight Center at JFK was designed by Eero Saarinen, architect of the Gateway Arch in St. Louis. The birdlike air terminal opened in 1962, but has been closed since TWA went out of business in 2001. You can view lots of photos of the interior by scrolling through the slideshow at the bottom of this article.
The two tubes you see emerging from the building in the bottom photo above are corridors that once connected to two satellite structures containing the boarding gates (1996 aerial photo). Those structures have since been demolished and replaced with a JetBlue terminal that opened in 2008 (2012 aerial photo).
Parts of the 2002 movie Catch Me If You Can were filmed inside the Flight Center; you can see the main terminal in this scene and one of the tubular corridors in this scene.
Ideas for reusing the terminal have come and gone over the years, but it sounds like there are now pretty firm plans in place to turn it into the centerpiece of a new hotel complex.
The longevity of the landmarked Flight Center is a testament to its architectural distinction, as JFK has not proven to be an easy place for notable but outdated structures to survive. The Flight Center's fallen brethren include Pan Am's Worldport, National Airlines' Sundrome, and the old American Airlines terminal whose facade featured what was once said to be the world's largest stained-glass installation (cool photo).
UPDATE: A few months after my visit, Governor Cuomo announced the approval of a long-term lease deal for the development of the site as a hotel.