To Link the Fire and The Dark Lord are the two endings that can be chosen in Dark Souls. The Chosen Undead may choose to perpetuate the Age of Fire, or instead bring about the Age of Dark. There are no other requirements, aside from defeating the final boss, to access either ending.
To Link the Fire
This ending is obtained by lighting the bonfire that appears after defeating Gwyn, Lord of Cinder. In sacrifice, the Chosen Undead links the First Flame as Gwyn had done before him/her, thus fulfilling Kingseeker Frampt's goal of continuing the Age of Fire. The Curse of the Undead will be broken, but one day the flames will fade again, and all will play out as before.
The Dark Lord
This ending is obtained after exiting the arena where the Chosen Undead felled Gwyn instead of lighting the bonfire. All the primordial serpents bow before the new ruler, the Chosen Undead, who has thus become the "Dark Lord". By choosing this ending, the Age of Fire is relinquished and Darkstalker Kaathe's goal of starting the Age of Dark is fulfilled.