She is the only figure in Norse literature who arranges the death of the man she loves, explains her reasoning at length, and is not condemned by the text for it.
Brynhildr's grievance is technical and absolute. She swore she would marry only the man who could ride through the fire around her hall. A man did. It was not the man she married. And the oath she made was not a preference but a binding formula, so the marriage that followed is void in the only sense she cares about — and the only remedy available is that the man who actually rode the fire must die.
Two women, or one?
The Poetic Edda presents a genuine textual problem here, not a subtlety of interpretation.
In Sigrdrífumál, Sigurðr climbs a mountain, finds a great light, and inside a shield-wall a sleeping figure in mail whose corselet seems grown into the flesh. He cuts it off with the sword Gram; she wakes, greets day and night and the Æsir, and names herself Sigrdrífa. She explains that Odin had promised victory to an old king, Hjálmgunnarr, and she gave it instead to his opponent Agnarr, and Odin pricked her with a sleep-thorn and decreed she should never again win a battle and should marry. She replied with a vow of her own: she would marry no man who knew fear. The rest of the poem is her teaching — victory-runes, ale-runes, birth-runes, wave-runes, speech-runes — followed by eleven pieces of ethical counsel.
Sigrdrífa means "driver of victory," which reads perfectly well as a title rather than a name. The Völsunga saga, working from a complete manuscript, treats Sigrdrífa and Brynhildr as the same woman without comment. Other Eddic poems and the German tradition keep them apart. The confusion is real, it is old, and it may go back to two separate legends — an awakened valkyrie and a foreign queen — merged before anything was written down.
The deception, and the quarrel at the river
Sigurðr comes to the Gjúkungs, is given a drink that erases his memory of her, and marries Guðrún. Her brother Gunnarr wants Brynhildr, cannot pass the flames, and swaps shapes with Sigurðr, who rides through in Gunnarr's likeness, stays three nights and lays a naked sword between them in the bed. The wedding takes place. Brynhildr believes she has married the bravest man alive.
The truth comes out in the least mythological way imaginable. The two women are washing their hair in a river and argue about whose husband is the greater, and how far upstream each is entitled to stand — because the water that runs off the better man's wife should not run down onto the other. Guðrún ends the argument by producing the ring Andvaranaut, which Sigurðr took from Brynhildr in Gunnarr's shape and gave to his own wife.
Brynhildr goes to bed, stops speaking, and does not get up. When Sigurðr comes to her and offers to leave Guðrún for her, she refuses: she will not have two husbands, and what she wants is not him but the undoing of a thing that cannot be undone. Then she tells Gunnarr that Sigurðr betrayed his trust on the ride — a charge she knows to be false, since the sword lay between them — and Gunnarr and Högni, oath-bound not to strike him themselves, put their youngest brother up to it.
The laugh, and the pyre
When Guðrún cries out over the body, Brynhildr laughs aloud, and Brot af Sigurðarkviðu has Gunnarr say the laugh comes from a heart cold with hatred. She then gives away her gold, foretells the entire subsequent ruin of the Gjúkungs in detail, and runs herself through with a sword. She asks that a pyre be built for them both, hung with shields and tapestries, with the sword laid between them again — this time in death — and with servants and hawks and hounds burnt alongside.
Helreið Brynhildar has her ride to Hel in a wagon and be challenged at the road by a giantess who calls her a husband-killer and says she should be spinning at home rather than seeking another woman's man. Brynhildr answers with her own account: that she was taken from her father, that Odin's sleep-thorn was unjust, and that she and Sigurðr will be together. It is a defence speech, delivered on the way to the underworld, and the poem gives her the last word.
What she is not
Wagner's Brünnhilde is a distinct creation: a daughter of Wotan, disobedient out of compassion, who returns the ring to the Rhine and ends the reign of the gods. That version has swallowed the medieval one almost entirely in popular memory — the horned helmet, the immolation, the redemptive arc are all his.
The Nibelungenlied's Brünhild is different again: a physically superhuman queen who throws spears and boulders, is overcome by trickery on her wedding night, and whose quarrel with Kriemhild is a dispute over precedence at a church door with no oath and no former love. She simply disappears from that poem after Siegfried's death.
The Norse Brynhildr is neither. She is a woman for whom a sworn thing is a fact about the world, who is deceived by other people's magic and other people's shape-changing, and who restores the only order available to her by destroying everyone involved, herself included. See Sigurd for the other half of the story, the valkyries for the office she was demoted from, and the Norse tradition overview for how the heroic legends relate to the mythological ones.
Was Brynhild a valkyrie or a mortal princess?
Both, depending which page you are on. Sigrdrífumál and Helreið Brynhildar have her assigning victory in battle at Odin's pleasure — an office, not a species. Völsunga saga also gives her a human father, Buðli, and a brother, Atli, which makes her the sister of the Attila-figure who later destroys her husband's killers. The tradition seems to have started with a supernatural figure and progressively naturalised her into the dynastic politics of the Burgundian legend, without ever removing the earlier layer.
Why is a sword laid between them in the bed?
It is a real legal-symbolic gesture, not a poetic flourish. Placing a naked blade between two people sharing a bed was understood in Germanic custom as evidence that no sexual act had occurred, and it appears in other medieval northern texts in the same function. Sigurðr does it to keep faith with Gunnarr while impersonating him — which is precisely what makes Brynhildr's later accusation a deliberate, knowing lie, and why the poems treat that lie as the gravest thing she does.