Ah that all important plot! The backbone of the story. For fanfiction the plot is slightly more flexible than in a novel or film script, as there is no set length. For our purposes this guide will go into the longer, more Novella-like structure, as opposed to the 'one shot'.
1) Keep characters true(ish) to the original story. Try to avoid letting the characters be too OOC (out of character). This kills the familiarity and the connection the reader has with the character. If you lose that connection, the reader may lose interest.
Example (Harry Potter): Draco Malfoy finds out that Harry is his brother so he forgives everything and they become best friends in a matter of two days.
Why it doesn't work: Mortal enemies don't become best friends overnight. If you really have to do this, make sure it takes a long long time and a lot of character development.
2) Avoid All Human AU. A popular idea that has arisen in fan fiction is to turn everyone in a book/film/anime into humans and putting them in a completely different setting, until all that remains of the original story is the names of the characters. Often so they can have sex. This is not fan fiction - this is using the names of someone elses characters in what is essentially a new literary universe. Doesn't make for good reading.
Example (James Cameron's Avatar): Jake can walk and is a business man, Neytiri is human and is his ditzy secretary. After many minutes of flirting Neytiri finally gives herself to Jake. They have a baby and call it Pandora and make Grace, Neytiri's best friend from Secretary School the godmother.
Why it doesn't work: This has absolutely nothing to do with the original story. The writer has taken away everything that fans of Avatar love about it, and stripped it down to a hollow shell that they've proceeded to fill with shallow characters that don't mean anything to anyone. The worst kind of fan fiction.
There are many more ways that plots can go wrong, these are just specifically meant for Fan Fiction. Most Fics that go down this route are pretty much guaranteed to be a train wreck.
(Harry Potter and Avatar belong to their respective owners...obviously)