To stay is to remain or wait in the same place. When you tell your dog to stay, you expect her to keep sitting until you call her over.
You can stay in the short term, like when you stay in your seat even though you're antsy, or in the long term, like when you decide to stay in Missouri after you graduate from college. You can also stay calm or stay out of trouble. Legally, to stay is to delay or suspend proceedings, like when a judge stays a decision. A less-common meaning of stay is "a support," from the Middle French estaie, "wood used as a support."