Simply put, a Traumatic C-Section is when a pregnant woman has her unborn child crudely ripped or cut from her body. If done well (being prone to Special Effects Failure) it can result in Grade A Nausea Fuel.
Contrast with Chest Burster, for when the offspring (or the equivalent of it) just violently comes out on its own, with no outside "assistance."
Often goes hand in hand with Eats Babies, for some reason. Clothes hangers and/or vacuum cleaners may or may not be involved.
A Spider-Man storyline involved the Venom symbiote being captured by a survivalist group, which forcibly extracted five "seeds" from it to create five new symbiotes for the their super soldier project.
Arguably the most infamous scene in The Anthropophagus Beast is Nikos ripping out a baby, and eating it in front of its dying mother. It's even worse in The Remake, due to the baby obviously being alive when Nikos takes a bite out of its head.
Automaton Transfusion re-creates the above scene almost exactly, but with a zombie.
In Las Vegas Bloodbath, Sam kills Barbara by cutting out her baby, which he casually throws at the wall afterward. It is later found crammed into a toilet.
The French film À l'intérieur ("Inside") centers on an insane woman who wants to cut out the main character's baby, intent on having it replace her own dead child.
In the otherwise forgettable made-for-TV film Scarlett, one scene features Scarlett O'Hara going through this.
During a test screening of the My Bloody Valentine 3 D, a viewer asked if it was Megan's unborn child and not her heart that the miner cut out, since the camera doesn't linger on what was ripped out for very long. It was her heart, and director Patrick Lussier mentions how squicked out he was by the baby assumption in the film's commentary.
Prometheus has a particularly scarring one. It ends up creating a squid-like Facehugger which grows to a giant size. Zoinks.
One of the many endless gore scenes in The Coma-Brutal Duel involves a ripped out fetus being stomped on.
The opening of Madness has one of the killers stomp on a pregnant woman's stomach so hard that what looks like her bloodied baby shoots out from between her legs.
This implied to have been done to the pregnant victim in Snuff 102.
The premise of The Clinic is that pregnant women are abducted by an underground organization, have their babies removed, and are made to kill each other in a twisted contest which prospective adopters use to judge which of the stolen babies they want.
In Orca The Killer Whale, the orca's mate is speared and taken on board the ship to be sent to an aquarium, but the resulting jolt causes her to miscarry. The whale fetus is then hosed off the deck, and the female is so upset by this she breaks free and commits suicide by swimming under the propeller.
Horrifically, this is actually Older Than Feudalism. It goes back to the root of all Western literature, The Iliad, in which Agamemnon scolds his brother Menelaos for showing mercy to a Trojan:
Agamemnon: Not a single one of them must escape sheer destruction at our hands. Not even if a mother carries one in her belly and he is male, not even he should escape.
Is mentioned in The Bible, specifically in the prophetic books of the Old Testament, where Israel has descended into idolatry and corruption, and is involved with conflicts involving other nations as well as its own civil war. The Assyrians (one of the most brutal and most effective armies of the ancient Middle East) are mentioned as doing this among other practices that would be considered war crimes today.
In the fourth Twilight novel, Edward rips open Bella's womb to retrieve their unborn child from her body with his teeth because both the nature of the pregnancy and the state of Bella's body didn't allow for any conventional birthing process to take place.
Fetuses in The Time Traveler's Wife time travel out of the womb on several occasions. This is extremely messy and disturbing.
A much milder example, in which the victim is a salmon, is mentioned in the Discworld novel Unseen Academicals. Still, it's only brought up for just how disturbing it was for young Vetinari to see all those cute little otter cubs eating all those little salmon eggs...
Vetinari: Mother and children, dining upon mother and children. And that's when I first learned about evil.
One of the tortures inflicted upon the female mutant in Apeshit.
From the 13th century Saga of the Volsungs: After seven years of pregnancy, King Rerir's wife orders her child be cut out of her body, even though she knows it will kill her. Thus, the hero Volsung is born.
Dantalion performs one on Amarantha after her carriage crashes in Literature/Maledicte to ensure the survival of an heir, even though the mother might live if he doesn't.
An incident from the childhood of Prince Joffrey in A Song of Ice and Fire is alluded to in Cersei's chapters which provides some early warning signs of his later psychopathy — while the details are sketchy, it's implied that he found a pregnant cat and cut her open to see what the kittens looked like.
In Metal Gear Solid 3: Snake Eater, The Boss, during the invasion of Normandy, went into labor. Naturally, being in the middle of a war zone without access to a proper medical facility, drastic measures needed to be taken. The most badass thing of all is that she performed the C-section on herself. Out pops baby Revolver Ocelot, in the middle of a battle, with his mother performing a C-section on herself.
Metamor City: this happens to Abbey Preston, by her abusive boyfriend. While she survives, she is extremely traumatised for years afterwards, especially as Sasha is killed at the same time. It gets worse - see, she's psychic, and it's hereditary, and she had even held simple conversations with her child in the womb. When her child died, she heard her scream.
SCP-1048 is an animate teddy bear that can make more of its kind using any random material, including a fetus it somehow removed from a dozing employee.
References to this practice in The Bible (mentioned above) are very much based on real-life Assyrian and Babylonian practice in war. There are poems dating back to 1100 BC boasting about the deeds of the mighty conqueror Tilgat-Pileser I, which included utterly destroying defeated peoples, down to their unborn children.
Casesarean sections had an extremely high mortality rate, and were thus almost exclusively reserved for those mothers who were already dead. Things only really changed in the late nineteenth century with the advent of (internal) surgery, something made possible by the invention of anaesthetics and disinfectants.
There's been numerous instances of this occurring in real life, with mentally unstable women usually being the perpetrators. Such cases likely inspired the aforementioned film Inside.
This was done to one of the victims of the Dnepropetrovsk Maniacs (the people responsible for "3guys1hammer").
The film section of this page references Black Sun: The Nanking Massacre, where a Japanese soldier uses a bayonet to rip an unborn fetus from a civilian woman's stomach. Unfortunately, this was based on an actual accounts by survivors of the siege.
Apocryphical accounts tell of a group of Child Soldiers in the Sierra Leonean civil war that bet on the sex of a fetus in a womans's womb, then cut open her belly and pull it out to check.
The only recorded case of a successful self-inflicted c-section could probably be considered a case of this. Inés Ramírez Pérez lived in a remote Mexican village when she went into labour. With no access to either a doctor or a midwife, no medical training at all, and unable to give birth herself, she eventually drank some tequila, took a kitchen knife and cut the baby out herself. Both mother and son survived.
Emergency c-sections can be this for those involved, as often the life of the mother, the child, or both is at risk and there isn't always time to be dainty. Before the advent of lasers for the operation, there were several reports of children born with cuts on their heads due to the surgeons slashing them with the scalpel in their rush to deliver the child .