Krang, the megalomaniacal evil brain from Dimension X, only existed in the animated cartoon version of Teenage Mutant Ninja Turtles, and the video games based on it. He appeared to be inspired by a race of aliens living on Earth from the original comic book series, although these aliens were not evil. Like Krang, they were brain-like creatures which had android bodies, which they controlled from the stomach cavity since they were too large to fit in the head. Unlike Krang's android body, however, these were designed to be covered with synthetic skin to appear human. No Krang-like entity made any appearances in the live action movies.
Krang's past was only vaguely hinted at in the series. In his home dimension, Dimension X, Krang was apparently some kind of mad scientist / warlord bent on conquering the entire dimension with his army of rock soldiers, armed with highly advanced technology. He was banished to Earth (by persons or forces unknown) for his crimes, and somehow lost his body in the process — being reduced to a semi-mobile brain which for some reason has short, prehensile medulla oblongata-like tentacles, eyes, and a mouth complete with teeth and apparently lungs because he can breathe and speak. One shudders to think what his original body must have been like for his brain to be so autonomous.
Upon arrival to Earth, Krang met Oroku Saki, the Shredder, and their common desire for power and conquest inspired them to join forces. Krang began building high tech weapons for Shredder to help him conquer Earth, in exchange for Shredder's promise to build Krang a new body so he could return to conquer Dimension X. Until then, Krang scooted around on a short, three-wheeled stool and later a small bipedal, chicken-legged walker with a glass enclosed top. Although he could crawl around on the ground, he was unbearably slow and left a trail of slime behind. Krang received his new android body in episode five, at the end of the first season, and was usually seen in it from that point on although he occasionally went back to the three-wheeled stool. I guess when he was having the android cleaned.
Like the comic book version android bodies, Krang's body was very humanlike. However the comic book's aliens designed them this way to disguise themselves as humans, while Krang's body was too bulky and robotic to pass for a human under any circumstances. Why it had a head and face was left unexplained, since Krang controlled it from the stomach cavity without any apparent way to access its eyes, if they were even functional. The seven foot tall android was a formidable battlesuit however, capable of transforming its arms and hands into weapons such as swords and chains. In its first appearance it was even able to grow about one hundred feet tall (with Krang growing with it) to rampage through New York, although Donatello smashed the device which made this possible by crawling inside of the giant robot with Leonardo. The growth device was never repaired.
As with any good cartoon mad scientist, Krang's inventions were often the driving force behind an episode's conflict. Krang built for Shredder an army of robotic ninjas called the Foot Clan which he used to start a crime spree in New York. Ostensibly human-looking, they wouldn't hold up to close scrutiny due to their hunched, bowlegged posture and elongated heads, although I assume they stuck to shadows like any good ninja would. The Foot Clan were far outclassed in their ninja prowess by the Turtles, however, mainly serving as expendable front-line troops for the Turtles to beat up during the series. Krang also provided dimensional warp generators, laser weapons, mutagenic ooze, and tunnel drilling vehicles to travel between the surface and their subterranean hideout.
And what a subterranean hideout it was! The Technodrome was a massive, mobile battle fortress which spent most of the series either without enough fuel to travel, stuck in a lava bed, or trapped in Dimension X. Many of Krang's plans during the show's run revolved around trying to get the Technodrome to the surface, where it could rampage unopposed by Earth's primitive military technology. Outside was a spherical shell running on massive treads, covered in weapons and nigh-impenetrable armor. Inside was a spacious high-tech headquarters with everything a mad scientist could want. The few times Krang managed to get it to the surface, however, he was stopped by the Teenage Mutant Ninja Turtles.