A self-insertion fan fiction is one which stars a fictionalized or fictional version of its author, or an avatar. These come in two major flavors.
The first type of self-insertion is one in which the avatar simply exists in the same fictional universe as the series in question, as an original character which "just happens" to strongly resemble the author. These are usually frowned upon, and closely associated with Mary Sues. It is often said that most fictional characters are at least in part their author; this type of self-insertion is an extreme version.
The second type is one in which the avatar is a resident of the "real" world, and is somehow transported to the fictional world. Usually, the avatar is a fan of the series set in the fictional world, like the author, and uses this knowledge to alter events. Sometimes, as in the case of Twisted Path, the avatar also gains significant powers and abilities in the process of insertion, especially if the canon characters are significantly more powerful than an ordinary human being. These stories are often frowned upon as well, but as the author is typically more self-aware than an author of the first type, some have nevertheless enjoyed moderate success. The labeling of such a story as a "self-insertion" is also much less subjective than the first type.
There is also a subtype of the second type, stories where the avatar possesses or otherwise takes the place of a character in the fictional series. Some stories have also been written which are essentially a form of the second type, with the variation that the protagonist is not an avatar of the author, but a wholly original creation.
-from fanfiction wikia, the fanfic wiki.
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