If my memory serves, Mary, Queen of Scots, and Bloody Mary were two different people. Bloody Mary was Elizabeth I's older half-sister. Her mom, Catherine of Aragon, had been put aside by Henry the VIII when he wanted to marry Anne Boleyn, Elizabeth's mother. Because the Pope refused to grant him a divorce, England broke with the Catholic church over that. Mary, the only surviving child of the first marriage, was traumatized by what her father had done to her mother (and her), and made it a point to try to bring back the Catholic faith during her reign. She persecuted Protestants, hence earned the name "Bloody Mary."
Mary, Queen of Scots was Elizabeth's cousin, I believe. She had a claim to the English throne herself (I've forgotten how), and I think Elizabeth may have ended by killing her to secure her own reign. But ultimately, the crown passed to Mary's son, James I of England (who was simultaneously James VI of Scotland). Elizabeth was a Tudor, but James was a Stuart (spelled Stewart in Scotland).