No is the ultimate negative: it means "not any," or "not at all," and it's also used to express a generally negative response. If there's no milk for your cereal in the morning, there's not a drop left.
A sign in a school hallway that says "No running" means that it's absolutely not allowed. And, the response, "No, thanks" when you offer your friend half of your liverwurst sandwich means that she is really not interested. You might vote no on a special ballot question about extending school to Saturdays or exclaim, "Oh no!" to express disappointment when your team loses a big game. No comes from the Old English negative, na.