A vestige is a trace or reminder of something that has disappeared or is disappearing. A lump of snow with a carrot in it might be the vestige of a snowman.
Vestige derives from the Latin vestigium, "footprint," and that is a good way to think of it — as a little hint of what was there before. It's something left behind, like a puddle after a heavy rainstorm. Sometimes vestige is used to emphasize a small amount; for example, you might say that the robber didn't show even a vestige of remorse after he was caught.