Staring Down Cthulhu

"But why--but why should any wish to kill me?" said Mowgli.
"Look at me," said Bagheera. And Mowgli looked at him steadily between the eyes. The big panther turned his head away in half a minute.

"That is why," he said, shifting his paw on the leaves. "Not even I can look thee between the eyes, and I was born among men, and I love thee, Little Brother. The others they hate thee because their eyes cannot meet thine; because thou art wise; because thou hast pulled out thorns from their feet--because thou art a man."

—Lampshaded in The Jungle Book

When a character is faced with a creature that should have him running away or cowering in fear—or which could at least finish him off with one bite—but instead beats it in an impromptu Staring Contest.

Picture the scenario: The Hero is cornered by the villain. Forget civility, reason, logic, God, this is serious. The Sealed Evil in a Can is about to be opened on the world, probably in the form of an Eldritch Abomination. The villain is locking the hero in a Cool Gate to Another Dimension with no chance of return. The Gladiator Games have commenced and it's one unarmed, wounded person against a whole pride of hungry lions. And there's nothing anyone can do. There's no backup, no escape, no Big Damn Heroes. All is lost, the world is undone, the lions are going to eat good tonight, etc. HE COMES.

But... wait. Despite having every reason to do so, The Hero hasn't turned her head in fear or closed her eyes to Face Death with Dignity. She just stares straight ahead. And she keeps staring, unaffected, unafraid; at the hungry lions, at the Complete Monster, even at that hideous, inhuman thing.

They keep staring. The seconds tick by. And the forces of evil start to squirm a little bit.

They lions don't move in for the kill. The villain stops his insane cackling. The Ultimate Evil... hesitates.

The hero, through sheer Badass, has intimidated something which had an infinitely superior advantage over him. Where the situation goes from here depends. Perhaps the bad guy turns tail and runs away. Maybe the hero's staredown has distracted the villain long enough for The Cavalry to arrive or the Big Damn Heroes to show up. Or maybe the villain gets over it and kills the hero after all, but they—and the audience—will remember that the hero may have died but he was never defeated.

A sure sign of the character being Badass. Overlaps with Death Glare. If the hero goes out staring down and intimidating his killer, it's a good candidate for a Dying Moment of Awesome.

Examples of Staring Down Cthulhu include:

Anime and Manga

Teen Genius Light Yagami of Death Note stares down Rem, a God of Death. Subverted in that the Death God is a nicer person than he is and through his manipulations he has turned her into an Unwitting Pawn.
A Death Glare is one form of "Haki" ("Will"/"Ambition"), a recognized superpower in the One Piece world. Can cause any animal or non-named character to faint.

Comic Books

Casanova Quinn is forced into a psychic staring contest with a hive mind of evil monks. Cass wins by having a mind full of spiders.

Film

This is how Mowgli defeats Shere Khan in the live-action The Jungle Book. Much is made of him being the one who can stare down the tiger and live.

Literature

The above scene from The Jungle Book (the book!), quite obviously.
Aragorn does this to Sauron with the Palantir in The Lord of the Rings... sort of. He challenges Sauron by staring into the Palantir and has enough willpower to break away from Sauron's visions.
Mouse, a dog, to Leanansidhe, one of the most powerful fae in The Dresden Files. Justified: Mouse is a Temple Dog, and this is exactly what Temple Dogs are bred for.
- In Proven Guilty, Harry himself does this to Maeve, the Winter Lady. Since she can squish him like a bug, it's a moment of understated awesome for Harry. Though, he does mention in his First-Person Smartass narration that he's really just up against a shark, and trying to make himself look nice and inedible and praying that the shark doesn't take a couple bites just to test.
And in a reversal of the above, the villain of Watership Down grants his troops seconds for a vital getaway by confronting a dog.
Ciaphas Cain does this to a regiment of battle hardened undisciplined soldiers fighting itself and any authority figure to stop them, with a handful of them using lethal force.

Live Action TV

Mr. Eko not only stares down, but actually scares off the Black Smoke, which is about as close to an Eldritch Abomination as Lost gets. However, in the next season, they meet again, and it kills him.
Criminal Minds: So, Hotch gets back to his apartment to find a serial killer waiting for him, a guy so bad he stabbed himself multiple times to throw police off his tracks. And Hotch shows no fear. Not when the guy shoots the wall right next to his head; not when the guy pulls out his knife, not even after the guy starts stabbing him! It's so badass, even the other characters speak in awe about how badass it is.

Video Games

In Final Fantasy I it is apparently possible to use a successful fear on CHAOS!

Web Comics

Bun-bun in Sluggy Freelance does this to an angry bear, making her (actually him, which may explain it) his crony. He's brought "her" cub along and everyone assumes he's going to go Mama Bear at the slightest provocation, but when Bun-bun isn't the least bit intimidated, he's apparently Genre Savvy enough to know a Killer Rabbit when he sees one.

Bear: "You're on your own!"
Cub: "Well $#!&."

Western Animation

Hercules, when riding CERBERUS to meet Hades in the underworld.
My Little Pony Friendship Is Magic: Fluttershy (pictured above) has subdued a dragon (who was so large each of his eyes was around the same size as her body) and a cockatrice (whose gaze was turning her to stone as she did so, but who still gave up first) by staring them down and scolding them.