Buck is the only son of Tallahassee and an unknown mother. His father is portrayed as very loving and caring, but unable to accept his loss. Tallahassee repeatedly refers to Buck as a dog early in the events of Zombieland.
Buck and his father made a duct tape wallet that appears as the only physical link Tallahassee retains to his dead son.
At some unknown point during the events of Zombieland, Buck was killed by zombies, an event which left Tallahassee depresssed and provides the impetus for the seeming vendetta he carries against the zombies.