Your bum is your bottom. This word can also be used to refer to a vagrant, although it's generally considered offensive or insensitive.
Bum is a name sometimes given to a beggar or vagrant: someone who tries to bum change from you. This word can also be used as a general insult, as in "The Yankees are bums!" (if you're a Red Sox fan) or "The Red Sox are bums!" (if you're a Yankees fan). A false accusation is a bum rap. As a verb, bum is a term for lazing around, as in, "I didn't have any homework this afternoon, so I just bummed around and played video games."