Your bum is your bottom, your rear-end, your behind. It can also be a vagrant or hobo, as well as a verb for lazing (bumming) around.
Bum is a name given to a beggar or homeless person: someone who tries to bum change from you. You don't have to be a bum to hum a cigarette from someone. Also, bum is a general insult, as in "The Yankees are bums!" A false accusation is a bum rap. If you remember that your bum is your butt — where you sit when you're being lazy — then the different meanings should fit together well.