He is an Australian brown snake who tries to eat Bernard (who falls in the water), Miss Bianca, and Jake, but the latter tames him not to. Jake in fact lassos the snake's nose and mouth with a rope, and gets him to take Jake and the other two mice with him to rescue Cody. Of course, while Jake and Bianca ride on his head, Bernard only rides farther away on his body, realizing that his love for Bianca was doomed. Twister isn't seen again for the rest of the film.
He bears some resemblance to Kaa from The Jungle Book. Unlike Kaa, though, Twister is perfectly harmless once you look him in the eye and let him know who's boss, according to Jake. Also, unlike Kaa, Twister has fangs.
In reality, snakes don't have eyelids, but the usage in animation gives them a more facial expression to the character.
Twister's name is only mentioned once in the film by Jake just before he lassos the reptile and says, "No mice for you, Twister, not today."