Police officers wear a badge — a shiny piece of metal that lets you know they're cops. Badges can also be metaphors — for example, a scar could considered be a badge of honor.
Police officers are the best-known badge-wearers, but lots of employees — especially federal employees — have to wear some kind of badge to work. The badge lets other people know who you are and what you do. Other kinds of badges are not so official. If a football player broke his leg during a big game, the cast could be called a "badge of honor," because of his playing and his toughness. All badges let others know who you are and why you're important.