Vhagar

"It wasn't just Aegon riding his dragon, it was Rhaenys and Visenya too...Rhaenys rode Meraxes, Visenya rode Vhagar."

- Arya Stark

Vhagar was the dragon ridden by Visenya Targaryen during the War of Conquest. Though smallest of the dragons, Vhagar was still large enough to swallow a horse whole, and its flames could melt armor.

Background
Vhagar was flown by Aegon's sister-wife Visenya during the War of Conquest. She was one of the three dragons present at the Field of Fire, the decisive battle of the conquest that ensured Targaryen mastery of the Seven Kingdoms.

Nonetheless, House Arryn thought that the Vale could still resist Targaryen domination by hiding behind the impregnable Mountains of the Moon along their borders, and fortifying the mountain passes beyond any hope of attack. Similar to how Aegon had dealt with Harrenhal, Visenya simply flew Vhagar directly over the mountains, and came to rest in the courtyard of the Arryn's own castle, the Eyrie. At the time, the King of Mountain and Vale was only a small boy, Ronnel Arryn, whose mother Sharra was acting as his regent. Visenya did not need to burn the Eyrie as Aegon had burned Harrenhal: her display of force with Vhagar was enough to compel the Arryns to surrender. Young Ronnel did ask one condition for his surrender: in return, Visenya would give him a ride around the Eyrie atop her dragon. The Vale surrendered, and true to her word, Visenya gave Ronnel a ride atop Vhagar.

Vhagar lived for another 130 years after the Conquest, until the great Targaryen civil war known as the Dance of the Dragons. In The Dance of Dragons: A True Telling, Grand Maester Munkun wrote that Ser Byron Swann wanted to kill the dragon Vhagar so he polished his shield for a week so that it was like a mirror and he crouched behind it and crept forward, hoping the dragon would only see its own reflection, but the dragon only saw a dumb man holding a mirrored shield and burnt him to a crisp.

Ultimately Vhagar died in the Dance over Harrenhal, in which she killed and was killed by the dragon Caraxes.

Season 2
While she is in the service of Tywin Lannister in Harrenhal, Arya Stark mentions that Vhagar was one of the three dragons used in the War of Conquest, ridden by Visenya Targaryen.

​Season 3
While incarcerated and learning to read, Davos Seaworth had trouble reading Vhagar's name.

Season 5
Shireen Baratheon is reading The Dance of the Dragons, A True Telling and retells the story of Ser Byron Swann to Ser Davos Seaworth which they share a laugh over.

In the Books
Laena Velaryon, daughter of Lord Corlys Velaryon and Princess Rhaenys Targaryen and wife to Prince Daemon Targaryen, rode Vhagar during the reign of Viserys I Targaryen. Laena eventually died in childbirth and not long after Vhagar accepted Prince Aemond Targaryen as his rider.

After Meraxes and then Balerion died, Vhagar was the last of the original three Targaryen dragons. Meraxes died fighting in Dorne, while Balerion later died of old age (about forty years before the Dance of the Dragons).

Vhagar was ridden by Aemond Targaryen during the Dance of the Dragons, the Targaryen civil war between Aegon II and Rhaenyra Targaryen. Aemond sided with his brother Aegon II. Vhagar was the greatest and most powerful living dragon during the Dance, by that time grown nearly as big as Balerion was during the War of Conquest, and she was one of the major advantages possessed by Aegon II's faction. In one of the opening moves of the war, Aemond rode Vhagar to attack and kill Rhaenyra's son Lucerys at Storm's End, while he was riding the young dragon Arrax - Vhagar killed both with little difficulty. At the Battle of Rook's Rest, Aegon II's forced laid a trap for Rhaenys Targaryen and her dragon Meleys (not the original Rhaenys, but her grandson Jaehaerys I's grandaughter). After Rhaenys arrived, she was ambushed by Aemond riding Vhagar and Aegon II riding Sunfyre. While Sunfyre was badly wounded, the great dragon Meleys was killed. Afterwards, Aemond used Vhagar to burn out vast swaths of the Riverlands, which had sided with Rhaenyra. Nonetheless, Vhagar and Aemond were eventually killed during the Dance, in a great duel with Prince Daemon Targaryen and his dragon Caraxes at Harrenhal.

She fought in more major Westerosi wars than any other Targaryen dragon: the War of Conquest, the Faith Militant uprising, and the Dance of the Dragons. Meraxes only fought in the War of Conquest. Balerion did fight in the Faith Militant uprising (ridden by Visenya's own son Maegor after Aegon died), but he died before the Dance of the Dragons - though Balerion did fight in previous wars outside of Westeros, such as the "Century of Blood" in the Free Cities.