In both the original comics and the 80's cartoon, Baxter Stockman first made his mark on the TMNT world by inventing the mousers – autonomous robotic rat-catching machines that couldn't resist trying to snack on the Teenage Mutant Ninja Turtles' sensei, Splinter. In the comics, he was secretly planning to use them to rob banks and collapse buildings (their jaws were strong enough to chew through concrete) despite the nearly guaranteed fame and fortune he could earn with them legally. In the cartoon, his inventions were rejected by the pest control industry but the Shredder took an interest in them as weapons. Another interesting difference was that he was a black man in the comics and white in the cartoon. Baxter Stockman does not appear in the live action movies.
The cartoon version of Baxter Stockman was a toady little nerd with a lab coat, bow tie, and squeaky voice. In the early episodes, he functioned as a mad scientist replacement for the Shredder while Krang was withholding his technological help as punishment for not defeating the Teenage Mutant Ninja Turtles. Eventually it became obvious that his inventions weren't doing the trick, and Shredder demanded that Krang send him the physically powerful, but bumbling and stupid mutants Rocksteady and Bebop, reasoning that "Maybe brawn will succeed where brains have failed."
The Technodrome was in Dimension X at the time, so a dimensional warp gate had to be opened to send the mutants to Earth. Krang chose a fly-infested garbage scow for the transfer point, causing Shredder to lament "I think he did it just to annoy me." In one of those desperate plot devices that only occur once in a cartoon's run, the Technodrome was so low on power that something had to be sent through the gate to Dimension X in order to send Rocksteady and Bebop to Earth, so Shredder threw Baxter, who had outlived his usefulness, through it to make the exchange. How the 98 pound Stockman was able to balance the half-ton of mutants sent through is beyond me.
In Dimension X, Baxter tried pleading for his life to Krang, offering his services as a scientist and inventor. Krang was unimpressed, however, being a scientist and inventor himself, and planned to execute the ineffectual genius in a disintegration chamber. Unfortunately for all involved, a fly followed Baxter through the portal from the garbage scow, and as we all know from the 1958 original and 1986 remake of The Fly, scientists trapped in molecule-scrambling machines with flies become hideous half human half insect monsters. Baxter's human form was never released as a toy.
Baxter the Fly gained fully functional wings, an extra pair of insect-like arms, and a fly's head with compound eyes. Baxter was the only recurring character who had the ability to fly under his own power, giving him a significant advantage over his enemies. His mind was somewhat deteriorated by the transformation however; not only did he develop an obsession with sugar and lose his attention span, but he also became rather easy to manipulate with a few smoothly spoken lies. Shredder was able to convince him that the Turtles were responsible for his transformation.
Throughout the rest of the series, Baxter the Fly was something of an unpredictable monkey wrench for both the Turtles and Shredder. Occasionally working with Shredder and even inventing one or two more things for him, Baxter often wound up fighting both the Turtles and Shredder after being betrayed yet again by the villain. Eventually he got it through his tiny fly brain that Shredder was not his friend and operated independently from there on out, his only ally being an alien computer named Z that, yes, was also manipulating him. Baxter's usual method of defeat was being thrown into some parallel dimension or another. Now I could understand once or twice, but this guy was thrown into alternate realities with a frequency unmatched by any other cartoon villain before or since.
I understand that in the new Teenage Mutant Ninja Turtles cartoon, which follows the comics much more closely, Baxter Stockman is once again a black man, and loses some body part in nearly every episode in which he appears. The last I heard, all there is left of him is a brain and one eyeball.