The Dance of the Dragons was a civil war in Westeros, fought between two branches of House Targaryen. It was a war of succession, fought between elder sister Rhaenyra Targaryen and younger brother Aegon Targaryen. It was conflict that lasted from early 129AC until halfway through 131AC. Thousands died.
King Viserys I had been married twice. From his first marriage, he had only one surviving child: Rhaenyra. Viserys had started educating her in ruling and had proclaimed her his heir. During his second marriage, to Alicent Hightower, Viserys had four more children: Aegon, Helaena, Aemond and Daeron. Though he now did have a son to succeed him, Viserys continued to insist Rhaenyra should ascend the throne upon his death.
When Viserys eventually died in the Red Keep in 129AC, Queen Alicent ordered the arrest of those few who knew of it, and plotted the coronation of Aegon with the small council, whilst Rhaenyra, on Dragonstone, remained unaware of her fathers death. Upon the discovery her younger brother had been crowned Aegon II, the war began.
The Dance of the Dragons was a war like no other. It was a war marked by stealth, betrayel and murder. Though savage battles were fought on the ground, much of the fighting occured on the water and in the air, because dragon fought dragon with tooth, claw and flame. The Targaryen dynasty survived and ended with the coronation of Aegon III, but the dragons were greatly reduced in number.