Stay means to be in one place or position for a period of time.
Stand means to be in an upright position on your feet. It can also talk about an object that is in an upright position without support.
- They had to stay in their car for three hours.
- With the help of coffee, she was able to stay awake.
- She had to stand in the queue for two hours to get the tickets.
- I was surprised the wardrobe was able to stand upright considering all the clothes in it.