If you rebuff someone's advances, it means they're trying to get in with you, and you're sending them signals that you are not interested. A rebuff can be quite rude, like pretending you don't hear their question.
Although rebuff is somewhat old fashioned in describing social relationships, it is still common in the world of diplomacy, where one country's rebuff might start a war, or end peace talks, or otherwise reverberate through the world of international relations.
