There are or have been at least three musical groups known as "The Story":
1) a UK psychedelic-folk duo
2) an American duo
3) a modern jazz band 1. The Story (UK)
The Story is a UK psychedelic-folk duo made up Martin Welham and his son Tom. They have released 3 albums - 2 on CD and the other on 12” vinyl. Their first release was a vinyl split with US alt-folk band Whysp in 2005 entitled ‘The Dawn Is Crowned’.

Read more about The Story on Last.fm.