The Pigeon-Rat was an experiment by Hugo Simpson, Bart's "evil" and formerly conjoined twin brother.
Hugo created the Pigeon-Rat by sewing the back of a pigeon to the back of a rat. The pigeon attempted to fly away through a broken window, but the rat on its back made it too big to fit through the opening, and it fell to the floor. Then the rat tried to run into a mouse hole, but the extra height from the pigeon prevented it from doing so.
Hugo made the Pigeon-Rat to practice his surgical skills, because he planned to sew himself to Bart, so they could be together again.