"Riding coattails" means using the fame or popularity of another person to increase one's own fame or popularity. Someone else does all the work, while you receive credit just by being seen with the hard working person. Can also be used in a business context. Ex.: "AMD (shares) riding Intel's coattails, but will it last?" This indicates, AMD shares are rising simply by being in the same sector (computer chips) as Intel when Intel's profits are up (but not AMD's, specifically).