The Brain was the first subject of the experiment that made Pinky and himself obsessed with taking over the world. The experiment worked in Brain's case, unlike Pinky's. He was made super-intelligent and the size of his brain, along with the rest of his head, was drastically increased. Every night, Brain comes up with a brilliant plan for world domination which is fool proof, but then Pinky comes into the picture. Brain usually ends up blaming the failure of his plan on Pinky even though it is neither of their faults.
Brain just has a lot of bad luck.
There is always one person involved with the plan who ends up foiling it. Sometimes it isn't even a person, it's a hamster named Snowball. Snowball grew up with Brain. They started out as friends, close friends. Then they were put through an experiment which injured both of them severely.