Other Names: OFC, OMC, Mary Sue, Marty Stu, Marty Sam, Gary Stu, Avatar
Location(s): If it can be found on a map (or even if it can't be found on a map), they've been there.
Dates: Post-Creation-the End of Time
There comes a point when the canonical characters of any fandom have told you, the author, all that they can about themselves and their world. A canonical can only be in so many places, view him or herself and his or her friends from a restricted number of viewpoints, and is simply incapable of experiencing all significant aspects of his or her world.
When an author reaches this point, the obvious solution is invent a new character, one who can, for example, tell about the events occuring in the Ethir when Aragorn and his band of merry dead men arrived, or give a less-than-charitable analysis of Éowyn's ride. Original characters can give depth to a story by providing the impression that a world does not consist solely of the chief protagonists and antagonists, and by helping to establish local traditions or to flesh them out in ways that the canonical text/medium and characters did not do. They can provide an outside view of the major characters that is unencumbered by personal acquaintance with the canonicals, or one which contrasts markedly from the views held of each other by canonicals. They can become your personal vehicle for exploring your chosen world
They can also be tricky to handle, acquire an unforeseen notoriety, and be hard sells to your reader. It's the reputation of some of their subspecies that casts suspicion on all OCs and inspires many a claim that requires instant qualification. Here are a few of the most well known species of original characters, some of which are almost solely responsible for reader distrust of their innocuous and even brilliant brethren:
OFC/OMC: Original Female/Male Character. In some fandoms, these terms appear to be used as the neutral, generic terms for original characters, with reference to their sex. Saying a character is an OFC or (more rarely) an OMC is not to immediately pass judgment on her or him. This has not proved to be the case, usually, with the species below. In practice in Tolkien fandom, it seems that the designation 'OFC/OMC' plays the role of both genus and the positive end of the continuum of characterization, which can lead to confusion as to which meaning is intended.
Mary Sue: The most infamous subspecies of OC. Some would say she's also quite inevitable. The definition of a Sue is not a closed topic. Some intend by "Mary Sue" to mean only an OFC; many, and arguably in Tolkien fandom most, readers and writers mean a badly conceived, clichéd character who does not seem to fit into the fabric of the canonical world, and who causes canonical characters to behave in ways that find little to no support in the original text/movie/medium. Paula Smith is the name you will most likely hear of online in connection with Mary Sue, as the person who wrote and identified Mary Sue as a character type in 1974. Pat Pflieger's essay is highly recommended as well.
Many have written of Mary Sue's defining characteristics, but the key to all definitions seems to be that whatever her features, she is the extreme example of them. If she is plain, she is extremely plain, the plainest Jane you will ever meet; if she is "temperamental" she is so temperamental she would blow the top off a thermometer by breathing on it; she is never merely cool, but induces frostbite with a glance; and if she is extraordinary, she is extraordinarily so. It is, however, more than Mary Sue's personal attributes that mark her out.
Arguably, it is context more than particular traits that determines Suehood, and determines it in a highly specific manner in fandom. The context of the story needs to support the OFC's character traits, however extreme. Within fandom, this generally means that the original environment(s) (physical, thematic, psychological) remain recognizable and that additions or explorations of said environment seem reasonable inferences given both the facts as written and the canon's overall 'look and feel.' Authors of Sues often do not change/deepen the context sufficiently to justify their specific characterization and plotline or else the altered context seems very arbitrarily altered solely in order to support the existence of the OFC in question.
Marty Stu: Mary Sue's brother has a number of names: Gary Stu, Marty Sam, Gary Sam. He seems to be less common, and more restricted in the behaviors he can adopt: more sterotypically masculine--the avenging, conquering hero or the dark and troubled rogue are two possible incarnations. But since he is essentially a male Mary Sue, he, too, will always be the extreme of whatever major character traits he possess, and his backstory will be at odds with reasonable interpretation of what the canon world will support.
Avatar: This type of OC is a thinly-veiled disguise for an autobiographical character. All characters of course have parts of the author written into them, even canonical characters. The difference is, again, degree and disguise. I am not certain whether there is a relationship between SI (self-insertion) characters and Avatars that is analogous to the OFC/Mary Sue relationship. Suffice it to say that, unless the plot makes it necessary that your character be known as your personal stand-in, the less obvious the Avatar, the less irritation s/he is likely to cause to readers.
It should be noted that canonical characters can also be characterized in such a way that they become Sues, Stus, or Avatars.
Mary Sue, Marty Stu, and Avatar OCs are usually viewed as being developmental stages--writers begin by writing them and gradually progress to original characters that are able to enhance a reader's enjoyment and understanding of a world. In this sense, they have an obvious function and even a necessary one which is entirely separate from the appraisal of the quality of their characterization.
Some Concluding Remarks
Original characters can be among the most rewarding characters to write because they are yours and you have nearly unlimited control over their destinies, unlike those of the established canonical characters. Some remain strictly background characters, props to serve a specific function: servant, messenger, soldier, carpenter, chandler, priest, or child. Others become minor characters, known by name and a line or two of dialogue, but not much else. Some become full-fledged characters in their own right. Although many readers may be suspicious at first, even an OFC deserves a chance, for "the world must be peopled," else it would not be a living world.
(Thanks to Stultiloquentia for reminding me of this particularly apt quotation from "Much Ado About Nothing.")