Baby Bowser is an antagonist in the Mario videogame series and made his first appearance in the 1995 game Super Mario World 2 for the SNES. He is a version of Bowser in his youth and is not to be confused with Bowser Jr., who in turn is Bowser's latest child.
All of Bowser's defining characteristics were already present in his baby form, albeit in a much less threatening way. At this young age, Baby Bowser was more of a bratty, spoiled child than an actual villain and the main motivations behind his actions are either self-preservation or just a general attempt to cause mischief. In his first appearance in the Mario universe, he was mostly driven by Kamek's prediction of the future, where he saw that Mario and Luigi will defeat him in the future (which became true in countless occasions).
Most of Bowser's defining abilities (namely the strength and fire-breath) are already apparent in Baby Bowser although in a much less developed way. However, with the help of his primary Magikoopa Kamek he is able to greatly enhance both his size and abilities. Of all the baby versions of the Mario series characters, his seems to be the most developed and is more of a small child than an actual baby. He was also already commanding a large number of minions, despite his young age.
Baby Bowser appeared in the Yoshi's Island games, as both an antagonist and teaming up with the protagonist, as well as Yoshi's Story where he again appeared as the primary antagonist. He also made an appearance in Mario & Luigi: Partners in Time, where he teamed up with his future self against the Mario brothers and their adult counterparts. As most Mario series characters, he also appeared in numerous sport and party games, as well as Brawl (although in most cases it is not clear whether it is Baby Bowser or Bowser Jr.)
In Yoshi's Story he appears on a floating creature that you have to knock down and after that you have to hit him until you beat him. Depending on which number or story path you choose will depend on how hard the boss will be. One is the easiest four is the hardest.