The House of Stuart (also known as Stewart) was a royal house of Scotland, England, Ireland and later Great Britain. The family name comes from the office of High Steward of Scotland, which over time became established as a family name.
The family were known as Stewart until Mary Queen of Scots, when she took the French spelling of the name, changing it to Stuart. There were nine Stuart/Stewart monarchs ruling Scotland until James VI also ascended the throne in England. The Stuarts then ruled both countries as a union until they were succeeded by the House of Hanover on the death of Queen Anne as per the Act of Settlement, which was brought in to prevent a Catholic monarchy.
Following the loss of the throne, the remaining descendants of James VII/II became known as the Jacobites and continued to attempt to reclaim the Scottish and English throne as the rightful heirs. However, there has been no more active claiments since the early 19th century. Later descendents of the Stuart family continued to claim the throne, despite the House of Hanover inheriting the throne. They were supported by the Jacobite movement, which you can find out more about here.