Your period doesn't always completely stop when you lay down however when you do lay down the blood flow decreases so you bleed less and are less likely to be having your period when you lay down.
Still you can still have your period even when laying down but your blood flow for your period is usually much less than it would be when you're sitting up or standing up.
When you're standing up, gravity helps blood flow out the vagina but when you're laying down blood doesn't flow out as easily, especially on lighter flow days.
But when your period is heavy blood flow then you'll still have a lot of blood flow when laying down so you should always wear a period pad when sleeping to absorb the blood that flows out.