Marshall Space Flight Center, Huntsville, Ala.
Not only would the "Eagle" not have landed on the moon without Marshall, no one would be on the International Space Station without the work of the center's scientists. They built and maintain the space station's oxygen generation and water recovery systems.
The center also lends NASA's telescopes a little X-ray vision. Marshall manages the Chandra X-ray Observatory, a telescope that records distant sources of X-rays in space. With it, we can see objects in space that would otherwise be obscured. The penetration Chandra gives to space is like the details doctors get when they use X-rays to see through the skin, down to the bones. Chandra has snapped the wreckage after galaxies collide and the mess left over after a star dies [source: Harvard].
Unfortunately for those who want to delve into the wonders of Chandra, Marshall has no official visitors' center. With a $15 to $20 admission, you can learn a little Marshall history inside of the U.S. Space and Rocket Center in Huntsville. But the main attraction there may be the rides that throw you around at several Gs.
Want to see NASA's mission control? Pack up and head southwest to the Longhorn state.