The National Aeronautics and Space Administration (NASA) is an agency of the United States government, responsible for the nation's public space program. NASA was established on July 29, 1958, by the National Aeronautics and Space Act.

http://en.wikipedia.org/wiki/NASA

And for NASA's version of it's birth: http://www.nasa.gov/mission_pages/exploration/whyweexplore/Why_We_29.html