Rainbow was formed in 1975 when Ritchie Blackmore got tired of Deep Purple's in-fighting, and he decided to quit the band to start a new project from scratch. Rainbow became more radio friendly and pop oriented than Purple would ever be, and that success kept Rainbow going for the better part of a decade.
Guitarist Ritchie Blackmore formed Rainbow after Deep Purple's Mark III version (which included himself, singer David Coverdale, and bassist Glenn Hughes) fell apart. Purple eventually re-grouped, but Blackmore chose to form his own band, Rainbow, with singer Ronnie James Dio. Dio had supported Purple as an opening act with his previous group, Elf. Nearly all of Elf became part of the initial version, called Blackmore's Rainbow.
The group saw success right away with their first single, "Man On The Silver Mountain" and would continue to find success while changing the lineup several times. Dio left after the third LP (Long Live Rock 'n' Roll) to join Black Sabbath after Ozzy was fired in 1980, and Cozy Powell, another key member, left to work with a number of others, including (some years later) Keith Emerson and Greg Lake.
In the end, Rainbow would have a revolving door type of membership, but eventually former Purple bassist Roger Glover joined up and American singer Joe Lynn Turner came in as the band's third lead vocalist in less than seven years.
While Purple often experimented with a soaring blend of British hard rock, neo-European classical, and American blues, Rainbow was a straight "corporate rock" band. Their signature style was complete with radio friendly hooks and the familiar Purple trademark sound that incorporated heavily amplified guitar and a prominent Hammond B3 organ played through a Leslie cabinet to give it a "swirling effect."