Fame is what you have if you're a celebrity: a lot of people know who you are. A musician's fame might mean she wears sunglasses and a baseball cap to disguise herself when she's in public.
Movie stars, rock stars, well-known public figures — these are all people who have achieved some amount of fame. You may have a kind of fame in a much smaller sphere: "She won every spelling bee in the state, which gave her some fame at her school." Fame is an Old French word that means "reputation or renown," from the Latin fama, "rumor, reputation, or renown," and also "ill-fame" or "scandal."