Bowser first meets Fawful in Mario & Luigi: Superstar Saga. Fawful quickly defeated Bowser when they confronted each other on the Koopa Cruiser. Since Captain Goomba gave Bowser some amnesia (revealed in the Superstar Saga remake), he didn't remember Fawful later, in Bowser's Inside Story. Soon he learns about Fawful's plots to rule all Kingdoms, and angered by this, strives to defeat the Beanish villain. He succeeds at the end of the game, defeating Dark Bowser as well.
In the main series, Mario and Bowser have had a long-lasting rivalry over who should be Princess Peach's partner.
In Mario Kart series, they get along and compete with other racers.