Dragonstone is a castle and an island off the east coast of Westeros that currently serves as the seat of House Baratheon of Dragonstone. It was one of the old seats of House Targaryen after the Doom of Valyria.
After the War of Conquest, Dragonstone was granted to the heir apparent to the Iron Throne to prepare them for ruling the Seven Kingdoms. These lords were known as the Princes of Dragonstone. This tradition was abolished after Robert's Rebellion, where the castle was permanently granted to Stannis Baratheon and his new cadet branch of House Baratheon.
During the imperialistic reign of the Valyrian Freehold and its Dragonlords, the vast Freehold colonized several islands of Blackwater Bay, most notably: Dragonstone, Driftmark, and Claw Isle. House Targaryen gained a fief on Dragonstone, House Velaryon on Driftmark, and House Celtigar on Claw Isle.
Twelve years before the Doom of Valyria, Daenys Targaryen had a prophetic dream where she saw the Doom of Valyria. She told her father Aenar about this, and Aenar fled with his family and household to Dragonstone to escape the inevitable doom that his daughter foresaw. The Dragonlords saw Aenar as a coward for this, and officially stripped House Targaryen of their legacy as Dragonlords. Aenar was famously known in Valyria as "Aenar the Exile".
Aenar would rule Dragonstone, and would then be succeeded by his son, Gaemon Targaryen, known as Gaemon the Glorious. Though the Targaryens brought five dragons with them to the island, four of them died, the survivor being Balerion the Black Dread. They had two dragon eggs, however, that hatched, giving birth to Vhagar and Meraxes.