To sully is to attack someone's good name and to try to ruin his reputation. If you spread false rumors that there's chicken stock in the vegetarian entree at Joe's Diner, you would sully Joe's good reputation.
Sully can also mean to tarnish or make spotty. It's easy to remember this meaning when you know that sully comes from the Middle French word souiller, meaning, "make dirty." For example, dripping chocolate sauce onto the table will sully your mother's new white tablecloth. Another meaning of sully is to corrupt or cast suspicion on. If an automaker recalls millions of vehicles due to safety problems, it doesn't exactly inspire confidence — in fact, it may sully their brand.