Spike is chasing a terrorist bomber named Ted Dampier (the "Teddy Bomber") who keeps blowing up buildings while giving dramatic speeches about his artistic philosophy that nobody listens to. During the hunt, Spike runs into Andy—a ridiculous wannabe cowboy bounty hunter who rides a horse, wears an enormous hat, and genuinely believes he's living in the Wild West. Andy immediately decides Spike is his rival and starts interfering with every bounty attempt, causing chaos and destruction everywhere they go.
The entire episode becomes a slapstick comedy of Spike trying to catch the bomber while Andy keeps crashing into him, through windows, with vehicles, and generally ruining everything. The Teddy Bomber keeps trying to explain his philosophy but gets completely ignored by both bounty hunters, who are too busy dealing with each other. Eventually the bomber gets caught, but Spike and Andy's rivalry remains unresolved—Andy still thinks they're destined nemeses while Spike just wants him to disappear. The crew back on the Bebop finds the whole situation hilarious.
See also: Session 17: Mushroom Samba, Spike Spiegel