The Age of Fire is an era in the lore of Dark Souls.
The Age of Fire began after the defeat of the Everlasting Dragons, at the hands of Gwyn and the other Lords. The Age of Fire was unnaturally prolonged when Gwyn sacrificed himself to kindle the First Flame.
When the Chosen Undead talks to Kingseeker Frampt, he will tell them that they are destined to succeed Gwyn. Should the player choose to kindle the First Flame upon the defeat of Gwyn, then they will sacrifice themselves as Gwyn did, thus delaying the Age of Dark once more.
Such a routine repeats itself for generations to come, until the time of Prince Lothric, when the prolonging of the fire proves futile.