Big Baby is a large, strong and merciless toy baby with childlike scribbles that resemble tattoos and a broken eye, made to look like a lazy eye. He doesn't speak much words, but rather gurgles.
In early life, Big Baby belonged to an owner named Daisy, along with Lotso and Chuckles. Then, one day, after a picnic, Daisy fell asleep and drove away, accidentally leaving Lotso, Big Baby, and Chuckles behind.
However, Lotso and the others went on their way back to Daisy's house, but it was too late when Lotso discovered that he was replaced with another Lots-o-Huggin' Bear. He took Big Baby away from Daisy's house, and he and Lotso took over Sunnyside and turned it into a prison. That said, Big Baby due to having the mind of an infant, wasn't aware that what he was doing was wrong.
Toy Story 3
Later on, when Woody and the gang were escaping Sunnyside, Lotso and Big Baby caught up with them, and Woody showed Lotso and Big Baby the locket that says "My heart belongs to Daisy". Big Baby started to cry, but Lotso didn't care and snatched the locket from Big Baby and smashed it with his cane, and abused by being called a "dummy", shocking everyone.
Big Baby, now fed up with Lotso's lies and personality, picked him up and tossed him into the dumpster, redeeming himself in the process. Then, he helped Woody and the gang escape when the garbage truck showed up. He also blows a raspberry at the tyrant.
During the credit roll, Big Baby is seen enjoying a happier life at Sunnyside, seemingly adopted by Barbie and Ken.
Big Baby watching a movie with his fellow 'Toy Story characters and other Pixar characters.
Big Baby in the credits.
The scene where Big Baby throws Lotso into the dumpster mimics the scene in Star Wars: Return of the Jedi, where Darth Vader throws Emperor Palpatine into the reactor shaft of the Second Death Star.
Because Lotso tricked him as well and took advantage of his young nature, Big Baby likely didn't have any agency over his actions. This makes him one of, if not the most sympathetic villains of the franchise.