Baby Luigi is Mario's brother Luigi, unsurprisingly in baby form. He was first introduced alongside Baby Mario in Super Mario World 2: Yoshi's Island, in which the aim of the game was to rescue him from Baby Bowser.
Yoshi's Island
In his first appearence, he was kidnapped by Kamek and taken to Baby Bowser whilst being delivered by the stork. It was up to Baby Mario and Yoshi to rescue him.
Mario & Luigi: Partners In TIme
Baby Luigi appeared in this game alongside Mario, Luigi and Baby Mario in which combined with Luigi could achieve combo attacks
Mario Kart Series
Baby Luigi first appeared as a playable character in Mario Kart: Double Dash!! as the partner of Baby Mario. His Car is the Green Pram, which is a lightweight car and his special item is the Chain Chomp He was absent from Mario Kart DS, but is confirmed to return in the new Mario Kart Wii. Adding to his stroller car, he also drives a motorbike
Continuity
Baby Luigi's first appearence was a change to the story of the Mario Bros. Although many games depict Luigi as the younger brother, Yoshi's Island seems to show that the brothers are in fact twins. This has not explained by any of the games