Dance of the Dragons
The Dance of the Dragons was the great Targaryen civil war fought over the Iron Throne roughly a century and a half after Aegon's Conquest. A war between rival branches of House Targaryen, it pitted dragon against dragon and all but destroyed the dragons of Westeros, marking the beginning of the end of Targaryen power.
Cause
The war broke out over the disputed succession of King Viserys I. He had named his daughter Rhaenyra his heir, but upon his death his second wife, Queen Alicent Hightower, and her faction crowned his eldest son, Aegon II, instead. The realm split between the supporters of Rhaenyra, called the blacks, and those of Aegon II, called the greens, named for the colors each wore at a long-ago tourney.
Dragon Against Dragon
The defining horror of the Dance was that both sides commanded dragons, and they turned these beasts against one another in the skies above Westeros. Battles such as the burning at the Gullet and the clash above the God's Eye, where Aemond One-Eye and Daemon Targaryen slew one another, saw dragons fall from the heavens. The slaughter of dragon by dragon devastated their numbers.
Aftermath
Both Rhaenyra and Aegon II perished in the war, Rhaenyra famously devoured by Aegon's dragon. The throne passed to Rhaenyra's young son, who reigned as Aegon III the Dragonbane. By the war's end nearly all the Targaryen dragons were dead, and within a generation the last of them died, ending the age of dragons. The full history is told in Fire & Blood.