Fafnir is not a dragon born a dragon. He begins the story as a dwarf — and his transformation is the story's entire point: greed, taken far enough, does not just corrupt a person's choices. In this telling, it physically remakes them into a monster.
The story begins with Loki, who accidentally kills an otter that is secretly the shapeshifted son of the dwarf Hreidmar. As compensation, Loki must fill the otter's skin with gold until it is completely covered — gold he obtains from another dwarf, Andvari, forced to surrender his hoard along with a ring, Andvaranaut. As Andvari hands it over, he curses the gold: it will bring ruin and death to whoever possesses it, a curse the rest of the saga takes entirely seriously.
Hreidmar receives the cursed payment — but his son Fafnir, consumed by greed for the hoard, murders his own father to claim it entirely for himself. Fafnir then withdraws to a place called Gnitaheath, transforms into a dragon, and lies upon the gold, sliding ever deeper into paranoid, isolated greed as the years pass.
Fafnir's own brother Regin, who received none of the cursed gold, raises the young hero Sigurd specifically to kill Fafnir and recover it — forging him the sword Gram from the shards of his father's broken blade. Following Odin's own disguised advice, Sigurd digs a pit directly in the path Fafnir regularly slithers along to drink, hides inside it, and stabs upward into the dragon's vulnerable underside as it passes overhead.
As Fafnir lies dying, he speaks — warning Sigurd that the gold's curse will follow him too, and that Regin, his own foster-father, is already planning to betray and kill him for it. Sigurd initially disregards the warning. But when Regin instructs him to roast Fafnir's heart and Sigurd burns his finger testing it, he instinctively sucks the burn — and tastes the dragon's blood, which grants him the sudden ability to understand the speech of birds nearby, who confirm exactly what Fafnir had warned him about. Sigurd kills Regin before Regin can kill him, taking the cursed treasure — and its curse — forward into the rest of his own tragic story.
Direct literary descendants: Fafnir is the clear ancestor of the dying, talking, treasure-obsessed dragon in Western fantasy — Richard Wagner adapted the story directly for his opera Siegfried (part of the Ring Cycle), and J.R.R. Tolkien explicitly named this material as a direct influence on Smaug in The Hobbit: the vulnerable soft spot, the fatal conversation with the hero, and greed made monstrous flesh all trace straight back to Fafnir.