Short answers, pulled from the story.
The House of Stuart was founded by Alan fitz Flaad, a Breton nobleman who served as the hereditary steward to the Bishop of Dol in the Duchy of Brittany. His son Walter fitz Alan became the first Lord High Steward of Scotland, establishing the family name from this administrative office.
The House of Stuart began ruling England in 1603 when James VI of Scotland became James I of England following the death of Queen Elizabeth I. This succession occurred through the bloodline of James VI, who inherited claims to the English throne through his grandmother Margaret Tudor, daughter of King Henry VII of England.
The execution of Charles I on the 30th of January 1649 led to the English Interregnum, an eleven-year period of republican government. Principal members of the House of Stuart lived in exile in mainland Europe until Charles II returned to Britain in 1660 to assume the three thrones.
The direct line of the House of Stuart ended in 1714 because Queen Anne died without living children after her sister Mary II died childless in 1694. The crown passed to the House of Hanover under the terms of the Act of Settlement 1701 and the Act of Security 1704.
The current Jacobite heir to the claims of the historical Stuart monarchs is Franz, Duke of Bavaria, of the House of Wittelsbach. The senior living member of the royal Stewart family descended in a legitimate male line from Robert II of Scotland is Andrew Richard Charles Stuart, 9th Earl Castle Stewart.