An Alternate Timeline or Elseworld (often in time travel stories) in which one small change has a ripple effect, resulting in massive changes. The technical term (from one of the seminal papers on chaos theory) is "sensitive dependence on initial conditions", but it's better known colloquially as "the butterfly effect". It's a Wonderful Plot is a subtrope where the protagonist is the nail. Butterfly of Doom is a subtrope where no good deed goes unpunished. The Unwitting Instigator of Doom is the character who steps on the butterfly/distracts the hero at the wrong moment/innocently reveals the Big Secret and will be hated for it.
Derives from a proverb-turned-poem which traces its origins as far back as the 14th century (making this Older Than Print). Probably the most well-known version is the one which appeared in Benjamin Franklin's Poor Richard's Almanac: 
For want of a nail, the shoe was lost; For want of a shoe, the horse was lost; For want of a horse, the rider was lost; For want of a rider, the message was lost; For want of the message, the battle was lost; For want of a battle, the kingdom was lost, And all for the want of a horseshoe nail.
A Staples Business Depot TV commercial from the 90s, directly references the trope with a different scenario.
For want of a paperclip, a page was lost. For want of a page, a contract was lost. For want of a contract, a deal was lost. For want of a deal, a factory was lost. For want of a factory, a company was lost. All for the want of a paperclip.
Toshiba Electronics commercials depict this inside the mind of the CEO, leading him to hold back on releasing the product to make sure it has all the features it needs, regardless of the cost. For example, adding shock-proof hard drives to their PC's. He imagines an electrician dropping his laptop, picking it up and plugging it into an outlet at a relay substation, which causes a spark that knocks out the power. This leads to a person's roommate opening the refrigerator, sniffing some milk, noticing it's spoiled but drinks it anyway, immediately turns into a zombie, and unleashes the Zombie Apocalypse on the world.
An ad series for DirecTV starts with a customer experiencing some flaw associated with digital cable, which cascades into random scenarios like waking up in a roadside ditch, selling your hair to a wig shop, or re-enacting scenes from Platoon with Charlie Sheen.
Anime & Manga
Higurashi No Naku Koro Ni has a bad case of this; every arc it's something else in the initial factors—minor things like a man moving into the village or even Keiichi giving someone a doll—that sets off a completely different scenario of bloody tragedy.
In Umineko no Naku Koro ni we have Lion, who appears in a fragment where Natsuhi accepted the baby that Kinzo gave her and didn't throw it over the cliff and therefore didn't grow up as Yasu. Though ultimately subverted in that Lion is still murdered during the massacre as Bernkastel gleefully shows.
A subversion is mentioned in the Ep 8 manga where it is mentioned that even if Battler would have kept his promise to Shannon/Yasu, Yasu would have still thought him/herself as furniture and so it still wouldn't have saved him/her
The basis for the plot of Dual! Parallel Trouble Adventure is that reality splits depending on whether a construction worker on site when Lost Technology is found takes a sample home when his crew's ordered to destroy the evidence. The Hero, Kazuki is at first able to see into the other reality, and eventually crosses over.
The reboot of the Rozen Maiden manga features Jun, the protagonist of the original manga, who decided to take the blue pill path instead, resulting in him not meeting the dolls and remaining a hikikomori for the remainder of his mid school duration. We meet him as a disgruntled college student with a bad side job, and frequently being laughed at due to his past status as a hikikomori. That is, until Shinku and the other dolls from the original continuity appear in his continuity...
One of the recurring themes in Legend of the Galactic Heroes, where seemingly minor events cause a huge impact on history. What if Kircheis had been armed during Ansbach's assassination attempt and survived instead of died? What if the supply depot Muller attacked had fought back instead of surrendering, so that he couldn't rescue Reinhard? What if Hilde hadn't persuaded Mittermeyer to attack Heinessen to force a ceasefire, or if Yang ignored said ceasefire? What if Yang hadn't allowed the "Imperial escorts" on his ship so that they couldn't assassinate him?
In the Mai-Otome manga, Sergay is not Nina's father and Nagi, while not a very good person, is not the main antagonist. As a result, Nina stays on the heroes' side for the entire story.
On a lesser scale, Sergay is not anonymously paying for Arika's education in the manga, forcing her to take on several side jobs and leaving her no time to study, at one point putting her in danger of failing and dropping out.
In Dragon Ball Z, the main timeline ends up being VERY different than the one Trunks comes from for reasons that aren't really well explained. For example, Androids #17 and #18 are much more powerful in the main timeline but not really all that evil (they do initially intend to kill Goku out of boredom but don't really go out of their way to hurt anyone else). Android #19 didn't exist in the Trunks' timeline nor did Android #20 (Dr. Gero had never turned himself into an Android). Goku also got sick MUCH later than he did in Trunks' timeline.
Actually much of this was explained. 17 and 18 were altered to be more powerful because of Trunks's arrival and the power increases of the other characters. Since Goku died in Trunks's timeline he saw no need and spent time making the androids into world domination machines (adjusting their personalities for the task) still strong enough to take on inexperienced Super Saiyans but still weaker than the main timeline versions. Gero probably altered himself and created Android 19 because of the appearance of more combatants with formidable power levels. This was because Trunks warned them about the androids and they trained to become more powerful. Also, in Trunks's timeline Goku was the one who fought Frieza on Earth by teleporting using instant transmission and the fight caused the virus to kill him sooner. Since he didn't go through any big fight, when Trunks killed Frieza, Goku never caused the virus's progress to accelerate and kill him when it did in Trunks's time.
The difference in strength between the main and alternate timeline versions of #17 & #18 was explained differently in the manga. In the ‘Trunks: The Story’ chapter when Gohan was fighting #17 & #18, immediately before #17 kills Gohan, #17 tells him that he’d been using no more than half of his power. After a short timeskip we see Bulma & Trunks debating the need for a time machine. Trunks thinks he’s already gotten strong enough to defeat 17 & 18, while Bulma thinks he’s too naïve since he was only at about Gohan’s level and Gohan was still killed. After this we see Trunks leave to fight them and the next panel we see they’ve put him in the hospital. There is no actual proof that the full magnitude of 17’s power (and possibly also 18’s) was revealed to Trunks before he went back in time. Assuming that Trunks was unaware of 17 & 18’s full power in his timeline, it would explain his surprise at their power in the main timeline. This would also indicate that the androids from the main timeline were actually just as strong as those in the alternate timelines and that Trunks was simply unaware of that fact.
In Blood+, Saya and Diva are twin sisters. Joel raised Saya like a daughter, whereas Amshel kept Diva locked in a tower, treating her like an experiment. When Saya receives this history lesson, she can't help but sympathize with her sister, knowing full well that they could just as easily wound up with their roles switched.
As revealed in Mobile Suit Gundam Unicorn, just about everything that's happened in the Universal Century, from the Earth Federation's MO to the One Year War and beyond, could be traced in some form or another to the Laplace incident in UC 0001.
The Walking Dead has a Fan Fic called Better Angels where Shane Walsh kills Rick Grimes during their confrontation (at the end of Season 2). This creates an obvious change in later events as Shane's leadership propels the group into The Prison.
Queen Serenity coughed while casting the resurrection spell that would revive her daughter along with everyone else. The result? The Moon Princess is Nanoha Takamachi, as seen in White Devil Of The Moon.
An exam proctor gets food poisoning results in Naruto becoming a ninja two years early in the First Try Series.
The Back to the Future fanfic "The Almanac Chronicles" has several, but one in particular stands out. President John F. Kennedy is not assassinated on November 22, 1963. It is implied this was due to Biff saving someone's life while on vacation in the Bahamas, a vacation he would not have taken had he not made his fortune from the almanac.
In the Total Drama story, Legacy, a delay of a few seconds has consequences that reverberate for years.
In the Superjail! fanfic An Unexpected Child, the Mistress becomes pregnant with the Warden's child after the events of her debut episode "Ladies' Night". This of course leads to a strange chain of events, involving a maze in the center of the Earth and a circus that gets its entertainment from slaughtering, that ultimately result in Mistress having the baby, a one-month premature little girl whom she decides to name after her mother and also Warden's mother in their honor, at the end.
A Minor Variation is another My Little Pony story that changes the events of the Sonic Rainboom, in this case by having rainbow dash literally lose a shoe for want of a nail. It diverges a LOT harder than Rainbooms and Royalty.
The A Song of Ice and Fire fanfic Ned Stark Lives is about Ned Stark, well, being saved from execution by Joffrey when Varys drugs him to prevent Littlefinger's plans of starting a catastrophic war come to fruition. Apart from Jon's and Dany's stories remaining relatively unchanged, the state of Westeros is, in comparison to what happens in the books, much better because of this.
The central premise of Lyrical Nanoha fanfic Game Theory is that Precia Testarossa is a little saner than she was in canon, and consequentially knows that being an outright Abusive Parent is a really bad way of controlling Fate. The result is that canon gets turned on it's ear, and Butterflies Of Doom spread even more changes.
Queen Of All Oni diverges from canon at the start of season 4, where, rather than releasing Tarakudo, Daolon Wong's improvised spell instead transforms Jade back into the Queen of the Shadowkhan. After this, the story transpires almost the same as canon (minus Jade as the Big Bad), but with a growing number of differences — Viper, El Toro and Paco have a larger role in events, for starters, and there's a secondary group of MacGuffins that wasn't present on the show.
Lola rennt (a.k.a. Run Lola Run): the wildly variant endings of Lola's twenty-minute Groundhog Day Loop can all be traced back to Lola's reaction to the boy with the dog on the staircase.
Sliding Doors has catching/not catching a train as the "nail" and ends with dramatically different versions of the protagonist.
The 'nail' moment was marked with music and later in the film when the protagonist had to make a decision the music was played again. Although we didn't follow this 'split' it implied that there were more points where a single decision would be important.
However, the event that happened in one case at the beginning of the film discovering her boyfriend's infidelity still happened in the other case at the end of the film, after many near misses, so on a broader timescale the system was not that sensitive to initial conditions.
Some events are constant between the two timelines. The system is still highly sensitive, though: in one of them, the protagonist dies.
In the movie Frequency, a son manages to send a message back in time and save his firefighter father's life. However, because of this change, his mother, who was a nurse, never left the hospital to make funeral arrangements and so was on duty to save the life of a man who should have died. This man turns out to be a serial killer who kills the mother and is now still at large in the present of the son.
The whole "Pottersville" sequence in It's a Wonderful Life is an example of this. George's having never been born turns out to have an indelible effect on the lives of pretty much everyone he's ever known in real life. (Of course, the whole idea of the movie is how much one man can make a difference.)
The Back to the Future film trilogy is a series of nails showing what would have happened if a certain event did or didn't occur:
In the first movie, Marty McFly's father was a wimp who couldn't stand up to the bully Biff Tannen, but after Marty changed events concerning his father in 1955, he becomes a more confident man who has Biff eating out of his hand, and Marty's family life has also improved.
In the second movie, Grey's Sports Almanac was the nail that, when old Biff from 2015 gave it young Biff in 1955, caused Hill Valley in 1985 to become Biff's lawless empire where he killed Marty's father and married Marty's mother, adopting her children in the process. With the nail removed, history reverted to how it originally unfolded in the first movie.
Also in the second movie, an automobile accident that would have happened in the third movie when Marty was racing with his friend Needles was the nail that caused Marty's future history to go down the toilet in 2015, eventually resulting in his being fired from the company he and Needles were working for. When the accident was avoided in the third movie, the YOU'RE FIRED message that Marty's girlfriend/future wife Jennifer Parker had received from the future was erased, meaning that their future could potentially become a better one.
Lampshaded by the scriptwriters in the Audio Commentary of the first Pirates of the Caribbean movie. The entire plot is basically kicked off by a corset being laced too tightly.
The Narada being sent back through time causes a divergence, creating the 2009 Star Trek reboot. James Kirk's father dies by suicide-charging his ship into it, James himself is born in space rather than Iowa, he and Spock do not meet on the best of terms, and Vulcan is completely destroyed.
Don't forget Starfleet technology being more advanced thanks to the scans of the Narada taken by the Kelvin. Then there's the destruction of an entire Klingon fleet, as well as a dozen newest Starfleet ships with the entire graduating class of the Academy.
Rules Lawyer and borderline Jerkass Genie Rumpelstiltskin offers Shrek a Deal with the Devil in Shrek Forever After: Shrek gets one day as a lawless ogre in exchange for one day from his past. Rumpelstiltskin takes the day Shrek was born, thus creating a world in which Shrek never existed. One of the biggest changes is the fate of Fiona, who without anyone to save her from the tower chose to Screw Destiny and simply escaped by herself.
The verses to "It's All Who You Know" by the Newsboys are two scenarios of this trope.
In the first verse, a slip-up in an actor's facelift leads to the failure of the show that would have saved a TV network.
In the second verse, an dogsled racer forgets to take a cough drop and loses his voice. He and his dogs freeze to death when "the sled got snowbound."
Tom Wait's song "Misery Is the River of the World" even includes a line from the trope namer.
For want of a bird The sky was lost For want of a nail A shoe was lost For want of a life A life was lost For want of a toy A child was lost
The video for "Has It Come To This?" by The Streets features a split-screen of two possible day-in-the-life situations of Mike Skinner.
"The Want of a Nail" by Todd Rundgren.
The whole point of Jay-Z's American Gangster is basically showing the mirror universe where Sean Carter kept on selling drugs instead of going into music.
Bowling For Soup's "Almost"'s verses each have chain of events that might have happened, if not for the first event in the chain only "almost" happening:
Verse 1: "I almost got drunk at school at 14 / and I almost made out with the Homecoming Queen / who almost went on to be Miss Texas / but lost to a slut with much bigger breastes / who almost dropped out to move to L.A. / and was almost famous for almost a day"
Verse 2: "I almost held up a grocery store / where I almost did five years and then seven more / 'cause I almost caught for a fight with a thug / who almost made off with a bunch of the drugs / that I almost got hooked on when you ran away / and I wish I could've had the nerve to ask you to stay"
Verse 3: "And I almost wrote a song about you today / but I tore it all up / and then I threw it away"
In Norse Mythology, Freyr is doomed to lose the fight to Surtur at Ragnarok because he gave his good sword away to Skirnir.
Play By Post Games
Fate Nuovo Guerra has at least two so far, being an Alternate Universe story and all. More obviously, the destruction of Fuyuki City caused the Einzberns to pack up and move operations to Lucca, where the next Holy Grail War takes place. Less obviously, the lack of a living Shirou results in Luvia getting killed while fighting a Dead Apostle Ancestor by herself, as well as the very existence of the Edelfelt Sisters Lumi and Lempi.
During Magic: The Gathering's Time Spiral block, the middle set (representing the present) featured "alternate universe" versions of certain cards, primarily reprinting older cards into other colors that would best represent them if various mechanics hadn't been assigned as they were. Keeping in the spirit of "What if?", the flavor text on a good portion of the cards give quick looks at an entire subplot that revolved around what would've happened if Mirri had slain Selenia before Crovax had a chance to, thus fulfilling the terms of a family curse, turning her into a vampire instead of him.
In a nice scenario for a Dr Who style RPG the player's TARDIS comes to rest in the car park of a film studio. This causes John Wayne to have to find another spot so he is late and angry for his meeting to discuss playing Genghis Khan. Instead Ronald Reagan gets the role and cancer inducing radiation dose. President Wayne's foreign policy isn't a great success either.
John Wayne was a very heavy smoker throughout his life (at one point up to six packs of cigarettes a day!) and in 1964 nearly died of lung cancer (one lung was removed and The Duke needed supplemental oxygen for the rest of his life). Any contamination he might have received from the set of The Conquerer was likely just a Wafer Thin Mint given what he'd already done to himself.
This is actually a means to become 'fragged' or disassociated from the right timeline in the time-travel RPG Continuum. You can destroy a time traveler by introducing a few of these little changes into his perceived timeline. "Replacing the nail" is a defense. Did an enemy time-traveler realize you sat in a coffee shop and read a book on Saturday, July 8th, 1995? He can frag you by doing something as simple as stealing the book out of your bag. You can reverse it by replacing a near-identical copy into your bag before "past you" notices the change. Screwing over a major historical event could bring down the wrath of any number of time travelers, all of whom find themselves affected by the change.
Erfworld Book 1 starts with this trope. The narrated opening scene shows the world's creators dropping one extra gem into a resource spot, which pays for an extra squad of units to go to field against Gobwin Knob's army, which is just enough to tip the balances in a heated battle, which means that Gobwin Knob's chief warlord gets an arrow between the eyes, which means that Gobwin Knob needs a new chief warlord before The Alliance shows up on their doorstep, which means that Wanda Firebaugh's bad day is about to get a lot worse. And then Parson blows up a volcano. Judging from the way the story is going, this was just the start.
In a time-travel example: the Schlock Mercenary time-travel plotline in Book 6 invokes this trope, but isn't itself an example.
In 8-Bit TheaterSarda in his anger sends White Mage into a pocket universe, only to discover years later that the pocket universe was actually our own universe before it started, and this made White Mage "The Creator". Sarda had planned to become "The Creator" but he arrives a few minutes too late.
Homestuck has innumerable offshoot timelines which all create elements necessary for the existence of the primary timeline. In one of those offshoot timelines, Karkat never runs the Mobius Double Reacharound virus. Without the virus (or some effect the virus had, such as possibly the deaths of the lusii), Gamzee does his From Nobody to Nightmare transformation much earlier than he was supposed to, and succeeds in murdering all of the trolls minus Aradia.
In General Protection Fault, the Nega-Verse is what would happen if Nick and Ki were evil and acted on Nick's fantasy of taking over the world that was discussed in a brief arc early on. There are smaller examples, such as Nega-Trudy being as selfish as Trudy Prime until her mother's request for her to finde her sister (something Trudy Prime didn't hear about until after her Evil Plan failed) changed her for the better.
Parodied in this strip of Tom The Dancing Bug. Percival Dunwoody wonders if he caused the people of 2012 to turn into fly-men when he stepped on a twig in the Paleozoic; it turns out that he actually did it by killing off all the humans in 2011 and injecting flies with human DNA.
ThisSubnormality strip. If you look closely enough, the last panel nearly spells out this trope.
Decades Of Darkness has president Thomas Jefferson dying of a heart attack. His bumbling successor James Madison manages to anger the New England states enough so they leave the union and defend themselves successfully in a war, splitting the USA forever. The slave states get the majority and transform the rest into an expansionist slave-holding empire.
Look To The West`s very different world (Steampunk revolutionary France, a divided India, French Australia, loyalist America) all stems from George II tripping on the carpet at his coronation in 1727.
Well Meet Again's truly crapsack world solely emanates from the butterflies of a letter lost in the mail...
The 4chan imageboard /co/ had a thread that speculated what would happen if Bruce Wayne's parents never died. The result? Bruce and a certain fellow named "Jack Napier" end up becoming best friends, Gotham City goes through a golden age, and the readers cried manly tears.
4chan's /tg/ board does this as well, some of the more recent ones, the crusades are won in the third crusade, the slew of changes involve a massive Kingdom of Jerusalem, The Library of Baghdad still stands, the reconquista never happened to its fullest extent, and the rising heresies move the reformation in a very different direction (Cathars, Flagellants and Waldensians rather than Anglicans, Calvinists, and Lutherans.) And these are the most normal things to come from it.
In the Chaos Timeline, Genghis Khan dying from a minor accident in 1200 prevents the unification of the Mongol tribes and their subsequent global conquests. The world becomes very different...
No Spanish Civil War In 1936 sees a (non-Stalinist) leftist Republican Spain not only survive the attempted coup in 1936 (with Franco critical to the thwarting of that plot), but become a junior partner of the Allies when World War 2 rolls along, having an anarchist (Buenaventura Durruti) for a president, and great advances in Spanish influence in the arts of the mid-20th Century, all because Casares answered a letter from Franco that didn't go answered in real life.
The Union And Liberty timeline begins with a small change in Andrew Jackson's 1828 campaign platform and changing a few words in a John Calhoun speech.
A More Personal Union begins when a young man recovers from a mid-autumn cold. The young man is the king of France, and the cold would have killed him; instead he lives for another seven years, and his continued reign brings very big changes to France.