The king dies and both sides crown themselves
Viserys dies in his bed; before the body is cold the greens crown Aegon II in the Dragonpit, and on Dragonstone the blacks crown Rhaenyra. Two monarchs, two courts, one throne, and ÔÇö for the first and last time in Westerosi history ÔÇö dragons on both sides of a civil war. The maesters name it the Dance, which is a very pretty word for what follows.
The fuller tale
Viserys died in his bed, in the night, having said nothing useful. What happened next took less than a day and decided everything. The queen's party did not send for Rhaenyra on Dragonstone. They shut the gates of the Red Keep, sat through a small hours council, and crowned Aegon in the Dragonpit in front of a summoned crowd before most of the realm knew the old king was dead.
One member of that council, the master of coin, said he wanted no part of it and was killed in the room. Another was sent to the black cells. The Grand Maester wrote to Dragonstone anyway, or so he later claimed. The speed is the tell: the greens knew perfectly well that a slow succession would go the other way, because the lords had sworn.
On Dragonstone, Rhaenyra was crowned with her father's own circlet before her own court, and the war had two kings, two courts and ÔÇö for the first and last time in the history of the world ÔÇö dragons on both sides of it. The maesters named it the Dance of the Dragons, which is a beautiful phrase for two years of relatives burning each other's children.
Two sides of the same oath
The blacks had the better claim by the king's own word and the lords' own oaths. The greens had the better claim by the precedent the realm had voted for at the Great Council of 101, and they had the treasury, the city, and the machinery of government ÔÇö which in the first year meant they could pay soldiers while the blacks could only frighten them.
Neither side was in the right, and neither side thought there was anything to discuss. That is the specific tragedy of the Dance: it is not a war between good and evil or even between two policies. It is a family unable to agree what a promise was worth, with the largest weapons in the world to hand.
Elsewhere in the wiki
 See the world as it stood in 129 AC What became of it  if an ending is written, it is here. Spoilers.
The chronicle records no ending here ÔÇö or none the maesters have set down.