A part of the Smithsonian Institution, located in Washington, D.C.. It's really, really cool. One of its main attractions is the amazing collection of aircraft and spacecraft that it has. For example, when you walk in the front door, directly in front of you is a replica the Wright Brothers' original airplane. Up and to your right is Chuck Yeager's supersonic aircraft, and Charles Lindbergh's The Spirit of Saint Louis. And right in front of you is a rock from the moon. The moon rock actually isn't that exciting, as everyone who comes in touches it. It's basically a polished, smooth piece of black rock.
Other rooms have rockets (American, Soviet, and German) and other cool planes. There's the backup version of the Skylab that you can go into. There's also the backup of one of the moon landers.
Of course, it being a museum, there are exhibits. They range from theoretical (How to Fly), to the historical (balloons and gliders), to the military (aircraft carriers and World War II), to the modern (the Space Shuttle and satellites). A lot of the exhibits are interactive, hands-on kinds of things.
Finally, the Air and Space Museum has a great IMAX theater. They usually show a range of movies about flight, some of which have some spectacular images of outer space or simulating actual flight.
There's also an above-average gift shop, and a below-average cafeteria.
Enjoy!