Baby Bowser is Bowser's younger self in the prequel games. He first appears as the final boss in Yoshi's Island.
During the game, Baby Bowser is taken care of by Kamek. He also sets out to find Baby Mario and tries to thwart Yoshi's attempt to save the baby.
He also turned Baby Bowser into super-huge Bowser. This was one of my favorite boss fights of all time.
Baby Bowser also made another appearance in Yoshi's Story, but I never made it far enough to meet him.
I do remember his role in Mario And Luigi: Partners in Time. He was pretty lame.
I liked him a lot more in Yoshi's Island DS.
His Japanese name is べビィクッパ (Bebi Kuppa) which means Baby Koopa. Koopa is Bowser's Japanese name.