Prof. Harold Hill: Now, Marcellus, I need some ideas if I'm gonna get your town out of the serious trouble it's in. Marcellus: River City ain't in any trouble. Prof. Harold Hill: We're gonna have to create some.
When one turns an unfortunate and unexpected set of events into a positive and/or fortuitous outcome.
Note that the Chinese word for "crisis" doesn't actually translate as "a dangerous opportunity", but that doesn't stop heroes and villains alike from seeing one when this trope is invoked.
A crisis occurs. It might be quite serious or be a relatively minor event, and it might even be deliberately manufactured by a character in order to bring about this trope. Either way, the crisis makes things harder, scarier or tougher for everyone except one or more characters or groups of characters to whom it's not necessarily a bad thing, and who seem to not only persevere in the face of the adversity, but outright triumph. Sure, they may have to make sacrifices too, but that's an acceptable loss. Because now they finally have the chance they've been waiting for. Rather than respond appropriately to the crisis and nothing more, the character or group decides to use the situation to their advantage in some way.
While this tactic is commonly used by the Big Bad or Magnificent Bastard, it can also be played positively. The idea that meeting and overcoming a crisis can pave the way to greatness is a pretty common one.
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This is the basis for a major plot in Watchmen. It was the one by Ozymandias.
The Vultures in The Phantom are a criminal organisation devoted to doing this trope in the most cynical way possible: they watch for other people's crises and show up to rob everyone when they're vulnerable.
In Astro City, the time-traveling villain Infidel used to go to various historical crisises and ask which of the victims were willing to survive impending doom as his slaves. As it turns out: a lot of them.
In the Star Wars prequel trilogy, Senator Palpatine uses fear of the Separatist movement to gain emergency powers ala Julius Caesar. Although he more or less created the Separatist movement in the first place.
In Super 8, the kids take advantage of a train crash and the soldiers in town in order to increase the "production values" of their film.
In Oliver Stone's W, Cheney and Rumsfeld view the aftermath of 9/11 and the subsequent War On Terror as a unique chance to end business left unfinished by Bush Sr.'s administration and start a policy of Empire-building in the Middle East.
In Frozen, Hans' original plan was to seduce Princess Anna, and then arrange an accident to eliminate Queen Elsa, so Anna would become Queen of Arendelle (implying that he would then have Anna eliminated so he would become king). When Elsa's ice powers were revealed to the public and Elsa accidentally froze Anna's heart, Hans chose to leave her to die, tell the cabinet that he and Anna said their vows before her untimely death, and have Elsa condemned for killing her sister.
Live Action TV
If anything happens in Star Trek, the Ferengi will attempt to carry out this trope.
Rule of Acquisition #35: War is good for business.note Rule of Acquisition #34, incidentally, is "Peace is good for business".
In Star Trek: The Next Generation this is a recurring theme with Starfleet Politics where more than once Picard clashes with ranking officers who are taking advantage of a current crisis to get away with their misdeeds. In the aftermath of Admiral Sati demonizing Simon Tarsis for being part human and part Romulan, Picard points out that people like her will be waiting for the right time to flourish, spreading fear and ignorance to keep themselves in power.
This is a main tactic of Petyr Baelish in A Song of Ice and Fire, by his own admission. He creates crises to upset the status quo and take advantage of opportunities that present themselves in the resulting chaos. The scale of this is hard to grasp in the earlier books, but it soon becomes apparent that his destabilising actions could be said to be responsible for the entire war, at least in aggregate.
Manga and Anime
In Code Geass, Lelouch accidentally orders Euphemia to commit genocide, and mere minutes later, grabbed the opportunity to use the incident to start a war.
As it turns out in the Soviet victory scenario in Red Alert, the entire Russian war effort was a plan by Kane to expand the USSR, then topple it, and use the ensuing chaos to strengthen the Brotherhood of Nod.
In the original Starcraft terran campaign, Mengsk uses the zerg invasion to help himself gain power.
In Girl Genius, the incapacitation of Baron Wulfenbach (and ensuing chaos throughout his empire) and emergence of a Heterodyne were entirely unexpected, but several factions lept on the opportunity to take advantage of them (including one group who'd been planning a rebellion for years, and were just forced to accelerate their schedule). Interestingly, just about everyone KNOWS this will happen.
In Worm, this is for the most part averted in relations between supervillains and superheroes. In the event of an S-class threat, it's common practice to band together and take it down, to the point that it's standard procedure for when the Endbringers-massive Kaiju-like creatures that have destroyed entire nations-attack. When Armsmaster sabotages this truce for a shot at personal glory, therefore, it is both a perfect example of this trope and a Moral Event Horizon.
The Legend of Korra: This is revealed to be the reason as to why Tarrlok is taking such extreme actions against the Equalists and unaffiliated non-benders. He's only using the Equalist crisis to further his own goals of cementing his power over Republic City, which is why he shoots down any other more reasonable actions to take against them.
Both Hitler and Mussolini rose because their respective countries were in shambles following WW1.
This is also how the revolution of 1917 came to pass - if not for World War I's devastating effects on the country, it would've been much harder - if not impossible - to pull off.
The British government got a lot of flack for "burying" the release of some bad news about the economy in the aftermath of a train crash.
Infamously paraphrased by Rahm Emanuel around the time Barack Obama was elected.
Naomi Klein's The Shock Doctrine: The Rise of Disaster Capitalism is a non-fiction doorstopper of a book about people doing precisely this. A sample quote:
Hurricane Katrina has just hit New Orleans, what do you do? If you said "privatize public schools", you might just be a conservative.
Honestly, this happens a lot in politics. Any sort of disaster, tragedy or public outrage will inevitably prompt opportunistic lawmakers to use it as a springboard for their pet legislation, regardless of whether or not it would, or even could, actually help fix the problem. Specifics are not necessary.
According to both myth and historical record, Cao Cao - the King of Wei most known for his starring role in the Romance of the Three Kingdoms - was assessed by a renowned sage, who was famous for always being able to tell someone's true potential. He stated that Cao Cao would be 'A decent administrator in peaceful times, or a blood-stained hero in chaotic times.' A few years later, the long period of civil war known as the Three Kingdoms Period got started, and Cao Cao lived up the prediction by turning the crisis into his opportunity - eventually outmaneuvering both Wu and Shu, and seizing control of all China with the formation of a new Dynasty (it didn't last more than three generations before his trusted strategist, Sima Yi, pulled an Evil Chancellor and seized power for himself - but by then, Cao Cao was several years in the grave, so you can hardly blame him for that).
After the Fire of London, that effectively destroyed the center of London, Christopher Wren (and scores of others) proposed new plans with wide streets making up simplistic shapes to create a bold looking city. In the end however it was decided that London should be rebuilt the way it way with winding small streets.