"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."
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. 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. 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.
And in the other corner, we see Alice The Hero, seconds away from complete and utter doom, standing resolutely with her eyes fixed directly on what is going to kill her. She stares straight ahead, and she keeps staring, unaffected, unafraid; at the hungry lions, at the Complete Monster, even at that hideous, inhuman thing.
The seconds tick by. The forces of evil start to squirm a little bit. The lions don't move in for the kill. The villain stops his insane cackling. The Ultimate Evil... hesitates. Alice, through sheer Badassery, has intimidated something which had an infinitely superior advantage over her.
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 Alice after all, but they — and the audience — will remember that she may have been killed but she was never cowed.
A sure sign of the character being Badass. Overlaps with Death Glare and is often portrayed as a Kubrick Stare. If the hero goes out staring down and intimidating his killer, it's a good candidate for a Dying Moment of Awesome.
Examples:Anime and Manga
On two occassions, the titular character of Naruto stared down the Kyuubi. The first time, Naruto was blackmailing it for chakra; the second time, he was telling it to stop nay-saying.
This is how the Fairy Tail team, battered and barely able to stand, defeats a still fresh and well-rested Sting in the last day of the Magic Games: they stare him down until he realizes beating them like this wouldn't help him in the slightest and surrenders.
The master of Did You Just Scam Cthulhu?, John Constantine the Hellblazer, does this when face-to-face with the most powerful beings in the universe, such as Satan and God. He stares them in the eye while puffing smoke from his cigarette. He even flips Satan a finger.
Victor von Doom is intimidated by precisely nothing. For instance, when he faced the Overmind, an alien being with the telepathic strength of billions of aliens, and which had already mind-controlled Reed Richards, he looked it right the eye and didn't back down. Or when the Purple Man helped him Take Over the World, then tried to threaten him. Doom took off his anti-mind control mask and looked Purple Man right in the eye.
When Superman/Batman go to Apokalips to save Supergirl. Batman threatens to blow up Apoklips if Darkseid doesn't give up Kara, even though the New God pummels him Batman still threatens him. Darkseid later concedes.
In the MLP:FiM fan fiction The Monster Mash tale Rarity: Blood and Water, Count Drache attempts to use his vampiric mesmerism on Fluttershy. She meets his gaze with her own and it's the vampire who looks away.
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.
George C Scott in the 1980 horror movie, The Changeling, has the nerve to glare at the vengeful ghost of a murdered child and yell: "Joseph, you Goddamn son of a bitch! What is it you want?" It works temporarily.
In Hannibal, Hannibal Lecter stares down a vicious Doberman and it curls up in the corner and whimpers.
V'Ger of Star Trek: The Motion Picture effortlessly destroys anything that affronts it—including a Klingon task force of three battlecruisers—on its way to Earth. Depending on the cut of the film you're watching, the energy field that V'Ger creates around itself has a diameter either greater than that of the solar system, or just greater than Earth's orbit around the sun. When it arrives at Earth, the machine casually prepares to obliterate the planet's biosphere. None of that stops Jim Kirk from playing chicken with V'Ger, and winning; making the leviathan stand down by threatening to not reveal who its creator is if it doesn't.
Mouse, a dog, gets a few of these in. Two of the most impressive times are once to Leanansidhe, one of the most powerful fae in The Dresden Files, and later to Nicodemus, a man with a literal Fallen Angel in his head. 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. Even better, he does this while calmly, quietly, tonelessly talking about how he killed Aurora, Maeve's peer and equal. He ends his little talk by holding a chair open for Maeve, and asking her to play nice, as he doesn't want a confrontation.
And in Cold Days, Harry stares down Mab, and threatens to have Demonreach imprison her. She actually seems to take the threat seriously.
And in a reversal of the above, the villain of Watership Down grants his troops seconds for a vital getaway by confronting a dog. (Though he didn't just stare, he also fought, as the dog later turns up with bites and scratches, drawing a remark from a human observer that it "must have gone for something that put up a fight.")
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: 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. Reid and Garcia muse maybe The Reaper let Hotch specifically because he didn't even blink.
The Reaper: Is this part of my profile? You can't show me fear?
Hotch: If you don't see fear maybe it's because I'm not afraid of you.
Hot Pie, of all people, does this in Game of Thrones to Gregor Clegane. Clegane has made it a habit of coming down to where prisoners are held each day and personally selecting one person to be questioned and tortured to death. Another prisoner tells Hot Pie he's managed to avoid being picked for torture by invoking this trope on Clegane. The next time Clegane comes down to pick a victim both the prisoner and Hot Pie enact this trope. Hot Pie doesn't get picked. The other guy does. Hot Pie then needs new pants.
Arya does this to Tywin Lannister. Tywin's death glares are known to scare everyone, and Arya — a little girl — stares him down.
In Jekyll, Hyde not only stares down a pride of lions but puts them under his mind-control.
In Lois and Clark, Lex Luthor is introduced staring down a venomous snake which slithers away in fear.
Gideon Jura, a Planeswalker from the Magic: The Gathering franchise, took up a staring contest with that universe's closest thing to Cthulu: Emrakul, the Aeons Torn. It happened at the end of the Battle of Fort Keff. Gideon, spotting the horizon darkening as if to herald the arrival of the great Emrakul, beckons the inhabitants of Fort Keff to flee. Gideon however remains to witness the Eldrazi first hand before Planeswalking away to Ravnica to seek help. This epic staredown is depicted in his first printing: Gideon Jura.
Librarians in Metro 2033 are on par with the vicious Demons, Moscow's apex predators, in terms of danger level, being tough-as-nails, rather fast, and surprisingly hard-hitters. Your best option is to stare them down until they lose interest (or get intimidated themselves) and leave, as Ulman tells you to. Oherwise, you're in for, if not death, at least a very hard fight.
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.
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. Subverted (twice), however, when she tries to stare down Discord — he only pretends to be horrified to mock her.
Truth in Television: Very few animals can outstare a human (with the exception of fish and reptiles, which have transparent eyelids). You can try this yourself with your dog or cat. However, staring down a large animal is not always smart, because prolonged eye contact is an aggressive signal, and some critters (notably bears, rhinos, and highly aggressive guard dogs) may take up the challenge and attack.
That pride of hungry lions example? It's been done by people outnumbered by the lions 5 to 1. Felines prefer to catch their food unawares, avoiding a straight-up confrontation, and to a cat, the target of a stare equals "prey." So if you catch a feline stalking you and stare back, it tends to chicken out.