"House Targaryen" is a featurette and part of "Great Houses" section of the Complete Guide to Westeros, a special feature in the Blu-ray of Season 1 of Game of Thrones. It is narrated by Harry Lloyd as Prince Viserys Targaryen.
Viserys explains that Targaryens are "blood of the dragon", descended from the nobility of ancient Valyria, a once mighty empire in Essos. When the Doom destroyed Valyria and its people, House Targaryen survived, having settled in the island fortress of Dragonstone years before. They remained there for a century, until the rise of Aegon the Conqueror.
Instead of attempting to take the lands of his ancestors, Aegon sailed west for the Seven Kingdoms along with his sister Rhaenys and Visenya. To keep their bloodline pure, Aegon took both of his sisters as wives, following the custom of their Valyrian ancestors. Together, they landed on the eastern coast of Westeros with their blazon: a three-headed dragon, representing Aegon and his sister-wives; their words: "Fire and Blood". While their host was small when compared to the armies of the Seven Kingdoms, Aegon and his sisters had a secret weapon: the last of the dragons.
Aegon conquered every kingdom except for Dorne, which bowed to Targaryen rule a century later, and had the swords of his enemies melted down by the fire of his dragon, Balerion the Black Dread, and forged into the Iron Throne. The capital city of King's Landing was built where Aegon and his sisters first came ashore. Aegon also ordered the construction of a royal castle, the Red Keep, on the city's highest hill.
For three hundred years the dynasty stood strong in the face of rebellion, civil war and plague. But the line of the "dragon kings" was broken when Viserys' father, Aerys Targaryen, the Second of His Name, was overthrown in rebellion. Aerys was betrayed and slaughtered by Ser Jaime Lannister, a member of his own Kingsguard. Aerys' son and heir, Rhaegar, perished on the field of battle, at the hands of Robert Baratheon, who claimed the Iron Throne for himself.
And so today, the sole survivors of the storied Targaryen dynasty are Viserys, rightful King of the Andals and ruler of the Seven Kingdoms, and his sister, Daenerys. They were spirited to the Free Cities of the east by loyalists. There, they have lived in exile ever since, dreaming of the day they will cross the Narrow Sea and take back their father's throne.