Baby Bowser is the much younger form of Mario's nemesis Bowser. While he looks almost the same as his future-son Bowser Jr., we see him for the first time when Mario is a baby and Yoshi is trying to save Mario's brother Luigi. Bowser's caretaker Kamek wanted both Mario and Luigi to be Bowser's friends, kinda sad if you think about it.
Even sadder is that Baby Bowser beats up Kamek and tries to ride the "green donkey". After a semi-tough fight, he is defeated.
But then Kamek uses magic to turn him giant. This is by far one of my favorite boss battles of all time. This was also one of the SNES's attempts to create "3D" before the N64, PS1, and Sega Saturn truly did it for home consoles.
He returns again in Yoshi's Story, but since I hated that game, I never made it that far.
He is playable in Yoshi's Island DS (my 2nd favorite in the series next to the first game) as he rides on Yoshi's back spitting fire. But he eventually betrays you...
Baby Bowser in Yoshi's New Island is much like the original game, but lamer.
Especially his giant form. I barely remember it going on.
Kinda of the same in Yoshi's Woolly World , but the yarn fight was certainly more fun than the 3DS's New Island.
His only real role outside the Yoshi's Island series is Mario and Luigi: Partners in Time for DS. Since Mario and Luigi meet their past infant selves, Bowser had too.