Where is Dragonstone?
Dragonstone is a castle sitting on an island at the entrance of Blackwater Bay, off the eastern coast of Westeros. Historically, it has been the seat of House Targaryen, and the heir to the throne is often called the "Prince of Dragonstone."
Who Lived in Dragonstone?
Quite a few notable characters have lived in Dragonstone over the centuries. As the seat of Targaryen heirs apparent, many Targaryen kings lived there before ascending to the Iron Throne. King Viserys himself was Prince of Dragonstone before becoming king, and his daughter, Rhaenyra, becomes the first Princess of Dragonstone. Viserys's brother, Daemon, also lives at the castle for a time, although he's never officially Prince of Dragonstone. It becomes Rhaenyra's seat of power during the civil war known as the Dance of the Dragons.
Dragonstone's Significance in "Game of Thrones"
Centuries later, after Robert Baratheon's rebellion, Dragonstone is taken by House Baratheon. Robert gives the castle to his brother, Stannis, to create a new branch of their family, House Baratheon of Dragonstone, and Stannis Baratheon lives there with his wife and daughter. After Stannis's defeat, the castle is left empty until Daenerys arrives and claims it as her seat; she has a connection to the castle, as she was actually born there.