Her son James took the English throne after the death of his cousin Elizabeth I. He became the 1st monarch to rule both Scotland and England and thus was the 1st in the house of Stuart to rule both. The house of Stuart lasted unitil the death of Queen Anne in 1714 when it went to her 2nd cousin George (he was James I's great-grandson) and then was known as house of Hanover. This caused considerable nastiness with some people (Scotts) who thought the crown should go to Anne's little brother (also called James) however he wouldn't give up being a Catholic (had to be a protestent monarch) it got so nasty uprisings known as the Jacobite uprisings took place, but failed.
The royal family today is decended from Mary, Queen of Scotts, ofcourse she has other decendents, some of which are more directly decended. If you google the house of Stuart you can learn more about them.