The Flame of Chaos is a force in Dark Souls.
The Flame of Chaos was created by the Witch of Izalith when she attempted to use her Lord Soul to recreate the First Flame. When she was unable to control it, she was consumed by it and transformed into the Bed of Chaos, mother of all Demons. Most of her children were similarly consumed and transformed into monstrosities; her daughter Quelana, however, escaped and used the Flame of Chaos to create the art of Pyromancy.
After the defeat of the Bed of Chaos at the hand of the Chosen Undead, the flames of Chaos persisted. It would eventually be discovered beneath Eleum Loyce, in the form of The Old Chaos. The Old Chaos would go on to consume many of Eleum Loyce's knights, turning them into Charred Loyce Knights, and eventually claiming the ruler of Eleum Loyce himself. The Ivory King would eventually be freed, at the bequest of of his oracle Alsanna, by the Bearer of the Curse.
Though it was not successful in replacing the First Flame, it functions similarly to it, as Demons can become powerful enough to "link" it and rekindle it when it dies out. The chosen one for this task was the Demon Prince, whose will is within all Demonkind, but his defeat at the hands of Prince Lorian of Lothric, and later the Ashen One, left the demon race without a champion, rendering them extinct.