The Lord of Dragonstone is the ruler of the stronghold of Dragonstone upon the island of Dragonstone in Blackwater Bay. The ruler commands the loyalty of several vassal houses established on the islands of Blackwater Bay.
Dragonstone is the ancestral seat of House Targaryen. It was the westernmost outpost of the Valyrian Empire. Prior to Robert's Rebellion, the island served as the seat of the heir apparent, known as the "Prince of Dragonstone". The two titles were often, though not always, synonymous. The first Prince of Dragonstone was Aenys I Targaryen, son of Aegon I . The last Prince of Dragonstone was Rhaegar Targaryen
Dragonstone was given to Stannis Baratheon by his brother, King Robert after he won the Iron Throne. Since Robert did not grant his brothers the title "Prince", Stannis was styled the Lord of Dragonstone instead. Its current lady is Queen Daenerys Targaryen.
After the Wars of Conquest, the ruler of the island and the castle of Dragonstone happened to be the heir prince of the Iron Throne, titled as Prince of Dragonstone, until the fall of the Targaryen house.
In the books
In the A Song of Ice and Fire novels, "Prince of Dragonstone" is the title of the heir apparent of the Seven Kingdoms during the Targaryen dynasty. Dragonstone became the seat of the heir once the Targaryens established King's Landing as the capital of the Seven Kingdoms. After the War of the Usurper, the title was not officially used. Daenerys Targaryen was styled "Princess of Dragonstone" by the handful of Targaryen supporters left, but after her brother's death claimed the Iron Throne instead and ignored the title. Stannis Baratheon became "Lord of Dragonstone".
In the time between the Doom of Valyria and Aegon's Conquest, the Targaryen rulers of the island used the title "Lord of Dragonstone".
The use of a specific princely title for the heir to the throne parallels real-world monarchies: the heir to the ruling monarch of the United Kingdom (and before that, the monarch of England), is granted the title "Prince of Wales", while the heir to the throne of Spain is traditionally the "Prince of Asturias".