Elsewhere Fic

There are two kinds of Elsewhere Fic. The first is clearly inspired by the canon, but happens entirely elsewhere. It contains none of the canon events or characters, often none of the locations, only Original Characters. So, there's very little chance of damaging continuity or having canon characters act Out Of Character. So no Kirk, no Mulder, no Joey Potter ... just original characters. Thus you might have (say) a Buffy fic dealing with a Slayer in Ancient Egypt or a Star Wars fic about a Wookiee Jedi in the Old Republic.

The second type takes place somewhat closer to canon. With a canon location, there will be canon characters doing canon things. They may appear in cameos and small roles, but the real story revolves around one or more Original Characters. Writers new to the Elsewhere Fic may turn this into a Mary Sue fic, with the canon characters existing only to heap praise on the OC, but better examples have the canon characters around only to give the OC their next mission, or to emphasise the universe the fic is set in, sitting around in bars and talking about their missions.

The result is something a bit like the relationship between the various Star Trek series: set in the same universe but telling a different story.

An Expanded Universe typically includes professional examples of this. Compare to Period Fic, which does sets fanfic in a different time. Contrast with Transplanted Character Fic which supposedly does focus on canon characters, but drops almost everything canon about them. Not to be confused with fic for St. Elsewhere.

Examples of Elsewhere Fic include:

Anime and Manga

Many Strike Witches doujinshi, such as "Witch in Africa" (considered to be "semi-canonical") involves Alternate History equivalents to other World War II campaigns in that setting, such as Northern Africa and the Suez Canal.
It's just as common for writers to create fanmade Pretty Cure continuities with their own characters, setting, and premise than it is to write fanfic based on the established seasons. The canon, what with its constant continuity reboots, lends itself well to this.
- Really, any metaseries with a flexible premise attracts this type of fanfic. Fanmade seasons of Super Sentai/Power Rangers, Kamen Rider, and the like are also rather common. See, for example, Power Rangers Take Flight, an Elsewhere Fic adaption of Choujin Sentai Jetman.
If shipping isn't involved, this may be the most common Yu-Gi-Oh! fanfic type.
- Ditto for Pokémon.
  - Well duh. Both that and Digimon both lend themselves well to this.
  - Not suprising considering how huge the worlds are for these series, it's easy to write when you have the world built for you but no one there.
  - The One Piece fanfiction community is also quite well known for this, with many fics being centered on the adventures of completely original crews.
  - Don't forget Naruto's setting. Even in canon while there are many ninjas, each one is unique in his or her own way.
Scoop and Dire Straits are examples of two Robotech first-person Elsewhere Fics of the second type, though the protagonists of both stories meet some of the canon characters at various points. Notably, neither of the protagonists of the stories are named.

Comic Books

Many fics set in The Crow's setting will have an original character resurrected by the title bird to seek vengeance for their loved ones, keeping with the original graphic novels' premise of doing the same.

Film

Lots of Lion King fanfic features an entirely OC cast, or canon characters in minor roles only. Stories set many generations before or after the events of the movies are common.
- And Disney did this their own selves in The Lion King 1 1/2.
In the Titanic fanfiction Come What May, the main characters are two Mary Sues who board the ship and fall in love with two of the Titanic's real officers, with characters and scenes from the movie appearing occasionally. There are a number of Titanic fanfics that work along these lines: sometimes the movie characters only make cameos, or sometimes a secondary character is the OC's love interest.
Into the Wild is an Avatar fanfic roughly set during Jake's training with the Na'vi. It primarily follows two OCs as they travel through Pandora and develop their characters. A few canon characters get cameos, but they don't affect the story much. It even goes for Doomed by Canon with gusto. It's a bit like that.
Star Wars: Knights of the Old Republic is basically an Elsewhere Fic (at least in relation to the Star Wars films) in video game form.

Literature

Conspiracy Theory, set in the Harry Potter universe, proposes a Grand Unified Theory of everything weird that happened in mid-20th-century America.
Harry Potter and the Man of Unknown, the very first Potter fanfic posted on Fanfiction.net back in 1999, is the second type. It's set almost entirely at another Wizarding School with its own faculty and students, but the canon characters pop up occasionally and an extended sequence towards the end takes place at Hogwarts. Harry himself is The Ghost in that the story pretty much revolves around him, but he just barely makes an actual appearance.
Popular in Warrior Cats fanfiction, with stories being set either before or after the established canon but in the same universe; also, fics in the same style in different locations, like America.
Hogwarts 1835 is a Harry Potter Fic taking place in, wouldn't you know it, 1835. With none of the canon characters, it focuses on the lives of 'normal' Hogwarts students.
Most Battle Royale fanfiction tends to be the first type, ignoring everything but the base premise - young people killing each other in an isolated location with a variety of weapons over a short time period. Besides some version of the explosive collars and danger zones, absolutely everything else, including the specific rules of the Program, are alterable by the writer. That said, the original characters/plotlines are considered archetypal. Survival of the Fittest and 72 Hours are examples of this.
Similar to the Battle Royale example, a lot Hunger Games fan fiction is this. We are told from the beginning that the basic premise (24 teenagers forced by a cruel government into a Deadly Game each year, with only one victor allowed to survive) has been going on for 73 years before the main characters really enter the story, so there is plenty of room for fan fiction. A less common type II variant involves a canon character as a mentor in the Games, trying to keep an original character tribute alive.
The Ollivander Children features very few of the original Harry Potter characters, mostly Tonks, Mr. Ollivander (natch), and Dolores Umbridge.
The Servants of Ungoliant, an Elsewhere Fic based on The Silmarillion, is set in the same world and time period as the original story. However, it occurs in the unused continent known as the Dark Land, rather than in the familiar lands of Middle-earth or Beleriand. In addition to this, only a few canon characters appear or are mentioned in the story, while the others are Canon Foreigners, Original Characters, or a mixture of the two. This story also recycles and utilizes various unused elements that were featured in The History of Middle Earth and various licensed Tolkien-related works.

Live Action TV

A Different Prison is set inside a Watcher prison in the Buffy Verse, an institution implied to exist but unvisited in canon.
Peter David's Star Trek: New Frontier novels are nearly a pure example of this, except that a few of his characters played minor roles in Star Trek: The Next Generation.
Other than Riker, Troi, and Tuvok in high positions, Star Trek Titan does this as well.
- And Melora, who previously appeared in an eponymous Deep Space Nine episode.
In fact, many novel-only spinoff series of Trek are like this. The I.K.S. Gorkon/Klingon Empire series, for example, is a completely new crew other than Klag (Riker's rival in "A Matter of Honor") and Rodek (formerly Kurn, given a new identity in Deep Space Nine's "Sons of Mogh"); and the Starfleet Corps of Engineers series is mostly new crew as well.
Another non-fanfic example: The Buffy the Vampire Slayer tabletop roleplaying game offers a number of these settings for original characters to play around in, including the modern-day American northwest, a post-apocalyptic cyberpunk setting and a vampire-infested old west.
Doctor Who used this trope a couple of times during the RTD era, filming an episode each series where the main cast had little more than Cameo roles in order to fit shooting fourteen episodes into a production schedule designed for thirteen. The first was essentially an Affectionate Parody of Doctor Who fandom, focusing on a group of people who bond over their past encounters with the Doctor and the dark turn their friendship takes when their group is taken over by an alien that absorbed humans. The second involved a photographer being sent cryptic messages in the present day from the Doctor (and later, her own friend) who were trapped in the past thanks to the Weeping Angels.
Fan fiction writer Mr. Chaos is famous for using this type of writing, especially the second type. Glimpses, GX: Hunters, and The Greatest Story Finally Told are all examples of this.
- His Heroes Fanfic even goes so far as to lampshade this with its title:ELSEWHERE. It has run as long as the Heroes tv show, includes a massive cast of original characters, and has featured cameos by almost the entire cast of Heroes (with come, such as Claire Bennett and Adam Monroe, revealed to be related to the main characters) , with the story itself referencing each episode the newest chapter goes along with.
Pisay Heroes occurs entirely in the Philippines, with original characters and scattered references to the original Company in Heroes.
The Gift is almost not even the first type of this for Quantum Leap, except it does (will?) mention the main OC tutoring a pre-teen genius with the last name of Beckett. Well, that and the whole premise aside from the Gender Bender stuff.

Radio

Big Finish Productions is the ... wait for it... Master of this trope, as along with their Doctor Who dramas, they also have a LOAD of dramas concerning other parts of the 'verse. BFP actually started out this way, as Virgin Publishing first entrusted them with the Bernice Summerfield series (itself mostly an Elsewhere Fic series based on a novel-only companion of the Doctor).

Tabletop Games

The entirety of BattleTech fiction is basically Elsewhere Fic, even the professional works; it's actually rare for the story to cover something from the tabletop or vice versa. The least Elsewhere Fic book of all 40 or 50-odd novels is Grave Covenant; two of its battles were included in The Dragon Roars amid another fifteen scenarios.
- Also the Warrior Trilogy by Michael Stackpole, which details the 4th Succession War, which was detaied in a number of sourcebooks.
Similar to the BattleTech example, pretty much all fiction set in the Warhammer 40,000 'verse, professionally published or not, falls into this category.

Theater

A professionally published example is Rosencrantz and Guildenstern Are Dead, which is about the eponymous characters running around having existential crises while all the bloody vengeance palace intrigue high drama of Hamlet is going on in the background.

Toys

Many, many, many Bionicle fanfics feature a team of OC Toa, on a different island somewhere on the planet. Whether or not they're fighting the Makuta as well varies, though.

Video Games

Bloody Way is of the type two variety, as it takes place in Santa Destroy, the setting of the first two games. Most of the characters and assassins are original, but the author has said that the original characters will make appearances.
Since Castlevania is a long series of stories about varying protagonists, many of whom are related, that attack the well-established villain's castle, it is trivially simple to create an original character (related to canon ones or not) in a new year and have them get involved. It's not too different from how the actual series is written.
A Rose And A Thorn 4 by Harley Quinn hyenaholic is an example of the second type of Elsewhere Fic. There are Canon Characters doing canon things, and it's set slap bang in the middle of the Sonic the Hedgehog Sonic Adventure 2 universe (or at least, 50 years before it)... but the real story revolves around two Original Characters.
Enemy of My Enemy does not feature any of the original Halo characters or settings, instead focusing on several Original Characters, both human and Sangheili as they struggle to defend the colony of Crassus against a Covenant Loyalist attack force.
Due to legal restrictions dealing with actor likenesses and Cryptic's policies about in game NPCs (No-go to avoid confusion with official storylines), this is often what player-created missions in Star Trek Online end up as. You can talk about things that happen and refer to events or people, but no established characters are allowed to appear in missions.

Web Comics

The Basalt City Chronicles (a fanfic of Gene Catlow) is of the second type, but much of it also takes place in the comic's original setting.
Project Future is made by a collaboration of Dan and Mab's Furry Adventures fans and takes place in the same world, but 150 years in the future and only a couple characters from the original comic have appeared, and usually just a panel or two.
- Granted, there is a lot of in-universe roleplaying on the DMFA forum, which is where most of Project Future's creators met.
Sburb Patch Notes is a type II of Homestuck solely due to its use of Andrew Hussie as the CEO of Skaianet Laboratories. As Skaianet Laboratories was never really expanded upon in Homestuck proper, it has become a small ball of fanon.

Western Animation

Another non-fanfic example: in addition to being a Deconstructive Parody of Jonny Quest, The Venture Brothers also takes place in the same continuity.
- It may not anymore. Johnny Quest has been quietly retconned as "Action Johnny", and Race Bannon is now simply "Red".
Paths Less Traveled By is a Transformers Animated fanfic set inside Cosmos' mind. Cosmos himself has no relation whatsoever to the characters in the cartoon save for waving his arms around in a crowd scene. It's generally considered AU, though, since the relationship between the characters (Cosmos and beachcomber) is not canon (the characters themselves, though, most definitely exist).
Villains is the Grand Finale of the Justice League cartoon (the one with Lex Luthor and Brainiac) through the eyes of the Flash's Rogue Gallery as they watch it in their bar.
Elsewhere Fics of My Little Pony Friendship Is Magic are very common. Much of the world so far is unexplored or unexplained (for example, is Equestria the whole world, or are there other kingdoms ruled by other species of talking animal, like the buffalo tribes seen in "Over a Barrel"?), and while many fics usually include Canon characters, ones that don't take advantage of this openness.
- The fanfic Fallout Equestria is notable in that it managed to spawn its own universe, with other authors building on its foundation and using many of its original concepts. While the story (and the sequels) are set two centuries later and focus on the OC protagonists, the original Mane six are not only notable historical figures whose actions were largely responsible for the current postapocalyptic state of Equestria, but actually have a clear and strong impact on the lives and motivations of the heroes, despite being generally very dead.
- On a less epic scale, there are also a lot of fics that are set within the show's canon, but deal primarily with background characters. An example is Lyra's Lousy Day, which deals with the experiences of a popular background pony during the events of "Swarm of the Century".
Operation:SIN is a very slim type II for Codename: Kids Next Door, with only 3 canon characters (Father and Cree Lincoln, and her sister Abigail Lincoln), making very brief appearances and few other canon characters being only mentioned. It mainly involves the 2 main OCs, for a a pair of teenaged girls who are former KND Operatives trying to elude the Decommissioning Squad. Only to be forced to join the Teen Ninjas for protection from them, much to their dismay, yet later are both rescued by, and decide to join an anti-decommissioning rebellion called the Kids Eternally There, who both their parents have been secretly working for. This Elsewhere Fic serves as a prequel to Captainwii's debut KND fanfic, Operation:ROGUE, which is currently being re-written.
Brazillian fanfic The Guardian (O Guardião) is a type I, based on Avatar: The Last Airbender. It has currently almost nothing to do with the actual show, aside from a few mentions to the Avatar himself and bending.