The First Flame is the fire that gave power to the Gods. It is mentioned in the name of the final area, Kiln of the First Flame.
The player's ultimate goal is to link the flame and restore power to it.
However, whether it is necessary to link the flame at all is up for debate. Throughout the series it has been hinted at that the "Undead Curse" is actually the normal state of humanity. And the existence of the sentient hollows of Londor brings into question whether hollowing is such a horrible fate as the Gods claim. It is possible that the First Flame is the true "curse" shackled on humanity, with the Gods cursing humans with mortality using their power of light in order to make them less of a threat.
In many areas in Dark Souls the sun is seen to be in a permanent eclipse, representing the First Flame going out. However, the sheer darkness of the Untended Graves suggests that perhaps it is not an eclipse at all, but the actual sun going out due to it being part of the First Flame - the sun itself being a creation of the gods.