A weapon used as a deterrent, to sway people into acting in a certain manner. This comes from the second moral of the original story of the Sword of Damocles, where "The value of the sword is not that it falls, but rather, that it hangs."
More often than not, the weapon is intended to have some type of height advantage, to more explicitly invoke this image. This can include launchable weapons that once off the ground serve that purpose.
The Kill Sat may easily be an example of this. The Weapon For Intimidation is a somewhat related, smaller scale version. See also Gunboat Diplomacy.
Examples:
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Nationwide Insurance has a commercial where a woman's insurance deductible hangs over her head as a giant rock in a net, threatening her much like the titular Sword. ("The World's Greatest Spokesperson in the World" is able to make it shrink with Nationwide's "vanishing deductible".)
Anime and Manga
In Code Geass, Schneizel's ultimate plan is to scare the world into peace by hanging a Kill Sat fortress armed with the Geassverse's equivalent of nukes over their heads. Appropriately enough, the fortress is actually named Damocles.
The power of Kira in Death Note, to instantly kill evil or unethical people (or anyone you want really), has limiting the world to good, obedient people as its intended consequence.
In One Piece, the protection or threat of of some of the Three Great Powers is used to control specific locations, Whitebeard's protection kept people from attacking even the weakest of his territories, Boa Hancock's membership in the Shichibukai protected her home of Amazon Lily, and the threat of a Marine Admiral, Buster Call, etc., kept certain laws and policies in place for the marines.
The Three Great Powers are more of a delicately balanced Cold War. The deterrent is their respective strengths. The Buster Call would fit with this trope more, though there is no one weapon in the series yet that keeps any sort of peace. On a smaller scale, things are protected using the threat of powerful individuals.
In Castle In The Sky, Muska takes control of Laputa for just this reason.
In Soul Eater, Arachnophobia get their hands on an Artifact of Doom named Brew, but it doesn't work (because Medusa's team stole the real one and gave them a fake). However, Arachne still realizes it to be useful because everyone thinks they have it, and thus can still threaten enemies into retreating.
Literal example - the first episode of K has two massive swords hanging in mid-air, one of which is referred to as the "Sword of Damocles".
Comic Books
In Watchmen, Nite-Owl's Archimedes seemed to be intended to serve this purpose, and was used this way when there were riots in the streets. Though, as it was his ship, it wasn't exactly taken seriously.
This is also Ozymandias's plan - by creating fear of an extraterrestial attack, he intends to scare the planet into world peace. It seems to work, but close examination reveals he launched his plan just as the United States and the Soviet Union were about to start peace accords anyway, meaning that he's scuttled real peace in favor of peace based on a false fear, liable to collapse if it's ever revealed... like what may be happening a few days after the last panel.
Film
In Like Flint. Gen. Carter schemes to load a space station with nuclear bombs instead of a weather laboratory (under the code name Project Damocles) and launch it into orbit around Earth.
One of the songs in The Rocky Horror Picture Show is titled "The Sword Of Damocles". In said song, Rocky sings about how he's just been born and how he can already fear that something is about to go terribly wrong.
Rocky: (Singing) The Sword Of Damocles is hanging over my head. And, I've got the feeling someone's gonna be cuttin' the thread.
In Star Wars, this is what Tarkin intended the Death Star and its superlaser to be used for: The threat of having your planet instantly destroyed would (presumably) curtail the ambitions of any seditious systems and work more efficiently than actually physically oppressing someone by force. The Expanded Universe gave this philosophy the name "The Tarkin Doctrine", and revealed that The Empire built an impractically large number of various planet-destroying devices in an attempt to implement it in practice. Since they all had a nasty tendency to get hijacked or blow up due to Rebel Scum, sabotage or other unfortunate circumstances, the doctrine was a resounding failure.
In Austin Powers: Goldmember, this is the intended effect of Preparation H.
The evil professor from Real Genius makes his genius students build him a laser (while his Butt Monkey student/valet works on the tracking system) for a space-laser version of one of these he promised to the military (for a supposedly obscene amount of money).
Tony Stark's exposition of his Jericho Missile in Iron Man might be a variation of this; a weapon that only needs to be fired once.
Tony Stark: They say that the best weapon is the one that you never have to fire. I respectfully disagree! I prefer the weapon you only have to fire once. That's how Dad did it, that's how America does it... and it's worked out pretty well so far. Find an excuse to let one of these off the chain, and I personally guarantee you the bad guys won't even want to come out of their caves.
In The Reveal at the end of The Day The Earth Stood Still, Klaatu claims that this is why his people built Gort and other robots like him; the robots were purposely programed to destroy Klaatu's planet if war ever started, preventing the species from ever doing so.
Literature
Robert A Heinlein's Between Planets. The Circum-Terra space station has hundreds of nuclear missiles, enough to destroy any military force that threatened Federation control of Earth.
Referred to in The Dresden Files as the "Doom of Damocles." The "Doom" is applied to those the White Council believes to be just shy of irredeemable, and can summed up as "One strike and you're out." With "out" meaning, "beheaded by the Wardens."
Ghost Story reveals that Harry himself served as this for Chicago with a lot of the supernatural community. They found him so terrifying that most just didn't bother showing up.
Referenced in The Salvation War, when referring to Uriel. Considering what Uriel is capable of, even with tinfoil hats and cruise missiles to stop him, this is not surprising.
Voldemort in the Harry Potter series uses the werewolf Fenrir Greyback this way: do what he says or he'll send Greyback after your kids. As Lupin (who was himself bitten by Greyback as a child) puts it "It's a threat that usually yields good results."
Mission of Honor: When Honor shows up at Haven to negotiate a peace treaty, her fleet of superdreadnoughts in orbit are referred to as "an infinitely polite sword of Damocles"
Live Action TV
Star Trek: The Original Series episode "Assignment Earth". The U.S. is about to launch an orbital nuclear warhead platform. Gary Seven's mission is to make it malfunction to scare other nations into not using them.
This was taken very literally in Power Rangers Zeo, where King Mondo dug up a "powerful but unstable" weapon called the Damacles Sword which he had buried centuries ago so he wouldn't be tempted to use it. (Apparently, the sword was a self-induced version of this Trope, which he broke after thinking he could defy it. As it turned out, he could not. Trying to fight the Rangers with it only resulted in him losing (and being the first Big Bad of the series to fall in combat to the Rangers, a very dubious honor).
Other
The Older Than Feudalismtrope namer, of course. As you know, he was a courtier at the court of the tyrant Dionysius II of Syracuse (Sicily, Italy). When he praised the tyrant for being so fortunate, Dionysius decided to give him a lesson and offered him his throne. Damocles accepted, but the tyrant hung a big sword above the throne, held only by a single hair (of a horse's tail). Soon, Damocles didn't think anymore that Dionysius was so fortunate...
Of course, the moral has been somewhat skewed over time. Nowadays the moral is that people can't live a happy life if they're in fear. Originally, the idea was a harsh lesson in "being the king isn't as great as it sounds", and the titular sword shows that his reign could fall at a moment's notice, especially in the time it took place.
Tabletop RPG
Dungeons & Dragons. The Forgotten Realms spell Blade of Doom causes a sword-like blade of force to come into existence over the target creature. When the target performs a specific act the blade drops onto the target, damaging it.
Video Games
In Sonic Adventure 2, the Eclipse Cannon was intended to serve as this. This is why it was only fired as an example. No point in Dr. Eggman ruling an empty planet, is there?
Metal Gear Solid Peace Walker pretty much runs on this trope; set during the Cold War, the game's villain creates a series of bi-pedal nuclear tanks that, following an initial nuclear attack to prove its capabilities, grants the closest thing to peace through mutually assured destruction.
In Front Mission Evolved the main enemy is revealed to be a terrorist group known as The Sword of Damocles and they plan to use a orbital laser to keep the nations of the world under their thumb
At the end of the Ancient China chapter in Live A Live, Odi Wang Lee reveals he always has a pair of assassins living behind his throne, with orders to attack whenever they see fit. They're strong enough to kill him, and thus he must always remain on guard - this is how he stays strong enough to be a powerful martial artist. They also serve as a last line of defense when the Xin Shan Quan master wipes out Lee's inner circle.
Soul Calibur IV features "Critical Finishes", One-Hit KillFinishing Moves that can only be used when your opponent's Soul meter is empty and yours is full. Since the Soul meter empties through blocking, they are clearly meant to discourage excessive blocking and are not meant to be seriously used.
Western Animation
A bit of Viewers Are Geniuses in Justice League: Project Cadmus employs "Damocles Class Missiles", which were used to fire directly into the space based Watchtower of the League to preemptively start a war between superhumans. The use of them is explicitly because Cadmus could not trust the league to continue to protect the people of the world.
In a different episode, Vandal Savage attempted to use a space based rail gun to aid his world domination.
In Justice League Crisis On Two Earths, the Crime Syndicate has one they intended to use to control their alternate earth. It was even named Project Damocles.
The Avengers: Earth's Mightiest Heroes villain Kang the Conquerer travelled from the future in his ship, which, in his bid to conquer present-day Earth, is referred to as Damocles Base.
Mutually assured destruction. Advances in weapons technology, such as nuclear weapons, were supposed to bring about an end to armed conflict in the traditional sense. It...well...KINDA worked...
in Star Wars A New Hope
