The Tudor Dynasty was a legacy started by Henry Tudor. He was known as Henry VII the King of England who was married to Elizabeth. Their son Arthur married Catherine but soon died. The brother Henry the VIII then married Catherine and became the next King. After Henry VIII was Edward VI of England, his son.
When Elizabeth of York died, was the time when the Tudor Dynasty was also over. She did not leave any successor to the throne. However, Stuart Dynasty started when James I became King of England.