The B-1 was first envisioned in the 1960's as a replacement to the B-52 Stratofortress. The intial B-1A was developed in the 1970's and 4 prototypes were developed. It was originally to be a high altitude bomber but that change and it became a low altitude penatrator.
With the defection of a MiG-25 in 1975, it was discovered that the Soviets had lookdown-shootdown capability in their radars. This meant that a low level penetrator would be vulnerable and the program was cancelled in 1977. However flight tests would continue.
With Reagan, the B-1 program was reinstated as the B-1B. There are some notable differences between the B-1A and the B-1B, the most noticable is the absense of a spine down the back of the plane. This gave the aircraft a much lower radar cross section.
Cost overruns meant that only 100 planes would be built. This meant that it would never quite replace the B-52 but supplement it.
During the 80's, it was mostly used as a nuclear bomber. This changed when it was converted to conventional use as well. It can carry more bombs than the B-52 (but still isn't as cool).