1. What is Plagiarism?
The dictionary defines plagiarism as the unauthorized use or close imitation of the language and thoughts of another author and the representation of them as one's own original work. Basically that means that plagiarism is considered theft or stealing. After all, taking someone's else hard work and claiming it as their own is stealing because they take something that didn't belong to them in the first place. It's the same as taking someone's ipod, or drawing or whatever and claiming it all as theirs.
2. What is Fan Fiction?
Fan Fiction is a piece of fiction written by fans for other fans about the fandom they love. That means that the stories they write are base on pre-existing (books, television shows, video games, films, cartoons and so on) material and they utilize the characters, situation, settings from that particular fandom to create their own fiction.
Fan fiction is never done for profit, only for the love of the fandom they write for. So many who write fan fiction know to put a disclaimer stating that they aren't the creator of the characters, settings and that the credit goes to the real creator.
However, fan fiction writers do have the right to claim any original characters, situations, settings...basically whatever that was added on and wasn't originally in the original material as their own. After all, they took the time to create something new, something interesting even when they are utilizing characters, setting and whatever else from a original source.
3. When Fan Fiction and Plagiarism Come Together:
Now some plagiarizers believe that just because it's “fan fiction” it's okay for them to take someone's else fan fiction and get credit for it. In their mind they believe that just because fan fiction is illegal and people can get sue for copyright infringement, that they can get away with plagiarizing someone's fiction.
That is true because fan fiction writer can't legally sue someone for plagiarizing their fan fiction since they themselves use from an original source. So when it comes to copyright law, the fan fiction author has no recourse. So from a purely legal standpoint, they are completely powerless to stop someone from plagiarizing their work.
However, even though they can't do anything legally and despite what plagiarist think, fan fiction authors can still take action against them. Everyone in the every Fan Fiction community knows that plagiarizing in fan fiction happens and when it occurs in their fan fiction community they do something about it.
It's taken seriously and the plagiarists are dealt with as soon as they are found out. Even when the plagiarist isn't in the same fan fiction community the real author, and sometimes their fan fiction community they are in, do whatever they can to assist in taking care of the situation and making sure justice has been served.
4. Examples and Forms of Plagiarism:
a. Word-for-word (Copy and Paste) Plagiarism — which is when a plagiarist copies the exact text word for word from a source. It could be anything that they copy (and paste) such as a story, a report, a article or any other kind of text. With word-for-word plagiarism there is no changing of sentences or words to make it seem like they did it themselves. Even then they sometimes don't copy everything but only take a couple of paragraphs directly from a source.
It's important that when writers do this, that they provide the proper acknowledgment. Quotation marks or footnotes which is needed when dealing with research papers, articles, reports, books, and school/work assignments.
b. Paraphrasing — which is a restatement of a text or passages, using other words. A writer would basically read from another source and then 'paraphrase' it by doing a condensed version of the text. They would explain or clarify it in their own words instead of doing it word-for-word.
It's important to paraphrase as much as possible and that it express a new idea. It can't just be a trivial rewording of the original material. If paraphrase is done incorrectly it becomes plagiarism. Also keep in mind that even paraphrased material need to be credited since they were base on an original source.
Original Text: Students frequently overuse direct quotation in taking notes, and as a result they overuse quotations in the final [research] paper. Probably only about 10% of your final manuscript should appear as directly quoted matter. Therefore, you should strive to limit the amount of exact transcribing of source materials while taking notes. Lester, James D. Writing Research Papers. 2nd ed. (1976): 46-47.
Plagiarized Version: Students often use too many direct quotations when they take notes, resulting in too many of them in the final research paper. In fact, probably only about 10% of the final copy should consist of directly quoted material. So it is important to limit the amount of source material copied while taking notes.
As you can see the plagiarized version is pretty much nearly the same with only a few changes to it. A correctly done paraphrase would be this...
Acceptable Paraphrasing: In research papers students often quote excessively, failing to keep quoted material down to a desirable level. Since the problem usually originates during note taking, it is essential to minimize the material recorded verbatim (Lester 46-47). 
Even though I haven't used a fan fiction as an example the concept is pretty much the same. The plagiarist sees a story they like and they copy and paste it. Before they post it online they try to change it a bit by paraphrasing it. Sometimes plagiarist do it but not always. They might just paraphrase only a part of it but leave everything else word-for-word. But we do know one thing and that is the results are always the same; they don't do it right. You can always tell that they didn't do a good job of changing it, especially when you compare it to the original source.
c. The Apt Phrase (or Apt Term) — If you use an author’s original and distinctive term or phrase without citation, it is plagiarism.
In Naruto, the character Naruto often ends his sentences with “dattebayo” or “(verb)-ttebayo”. It's is an unique style of speech that only Naruto has and in which only the creator, Masashi Kishimoto has made up for that character.
It would be plagiarism if one takes that same phrase for their own original character without the proper credit. Even in fan fiction, authors can come up with their own distinctive apt phrase that no other fan fiction author would have thought up. Such as if they have the characters saying something like, “We're awesome and together our awesomeness shines brightly like a lighthouse in a foggy night.”
It's very unlikely that someone else would ever come up with something like that. Even using a single word without acknowledgment can be apt phrase plagiarism, if the word was coined by the source from which is taken. Such as the usage of the word “awesomeness”, or even a made up word.
Apt phrase is like a fingerprint and relies on the author's mind and style of writing so each person's writing is different and none is alike.
Fan fiction is base on something already created but what's unique about it is the plot ideas, added characters, situations, the style it's written in is new and different. That's all created by the fan fiction author so no matter how many times a plagiarist tries to change it so that it doesn't look the same, there are still remnant of what it was before. That's also a type of plagiarism and it's called...
d. Mosaic (Word Switch) Plagiarism — which is language closely based on a source but with a few words changed around. It's the opposite of word-for-word where the plagiarist change words and other things to make it seem like they came up with it and they try to make it as though they wrote it. Now if this was a research paper and they were using a quote and changing words, it wouldn't be plagiarism if they use the exact quote (with quotation marks and citation), or express the point in their own words and cite the source for the idea.
e. Idea Plagiarism — is when the author of the source material expresses a creative idea or suggests a solution to a problem, the idea or solution must be clearly attributed to the author. That means if someone came up with something original that hasn't been written (or thought) about, another person can't take their idea without giving them credit for it. This is where one has to distinguish the difference between an author's ideas/solutions from common knowledge.
In fan fiction it is common knowledge even if it's a cliché, that when a being (be it demon, sayian, werewolves or some other creature) bites another on the neck they are “mates” for life. That's common knowledge in fan fiction and doesn't need to be reference.
Just as it's common knowledge that the sun is a star, or that chickens (the hens of course) lay eggs.
However, when something isn't common knowledge it's important to give credit of where the idea/solution came from.
If a fan fiction author came up with the idea of tattoos between demons to mean they are married/soul-mates and created a spell and/or a rare flower or fruit to make the ink then that idea is theirs. It's something that has never been heard of or done before and if one plans to use it they must ask the author's permission to use it AND credit them.
If for some reason one can't ask then at least credit them and let people know where to find them. Plus, just because one borrows the idea and alters it to fit their story/plot/characters or whatnot, it doesn't make it yours either.
Remember the idea originally came from them and you still have to give them credit. After all, if their original idea inspired yours then it's only right to give them kudos for it. If they didn't have that idea then you wouldn't have yours either.
5. Why would someone plagiarize in the first place?
There aren’t many reasons I can think of but most of them are because of selfish reasons. Mostly it’s because they are too lazy to write something themselves and just don’t want to do it. They want to experience that instant fame. They don’t want to work hard writing stories and they want to be the best. So they do that by stealing from the best and they enjoy the attention they get from those “stories of theirs”.
It’s quick and easy for them and so they do it believing they won’t get caught.
6. How do you know if your story has been taken?
The internet is a huge world and plagiarists rely upon the anonymity and massive amount of space to hide their activities. They know what they are doing is wrong but they still do so because they hope that they won't get caught. So they look for stories that do well with readers/reviewers and takes those stories by posting them somewhere that no one has ever heard of but still gets a decent amount of traffic.
Even though they put in the effort to hide what they did, they don't realize that the same tool that makes it easy for them to find works to steal can also make it easy for anyone (especially the original author) to hunt them down. Because, even though the Internet is vast, it’s so well indexed that finding plagiarism is a very easy task.
Google is the biggest search engine and all someone has to do is input something from their writing and search for it. I suggest not searching for the title for your story since plagiarist aren't stupid enough to use the title of your work (but there have been some cases in which they were). Some do change the title of your work while keeping the body intact.
So what helps the most is putting in the search engine something from the body of the story. It must be something unique to your work and uncommon.
What if somewhere in your story you have this...
The cat woman stared as the human man hanged from the tree, his boot clicking together and making the only noise in the eerily silent night.
It's not every day one sees or hears this, let alone read something like this in a story. So copy and paste that sentence and search for it in Google. If the results is too broad and you want to narrow it down then try putting it in quotes. This way the exact phrase will appear and you'll have a better chance at hunting down your story if it was plagiarized.
If it doesn't help then try putting key words in quotes. Also if your story is about an anime, for example; Yu Yu Hakusho or anything else (even canon characters) then put it in your search too. So your search engine should look something like this...
The “cat woman” stared as the “human man” hanged from the tree, his boot clicking together and making the only noise in the “eerily silent night” Yu Yu Hakusho.
You can even remove Yu Yu Hakusho with a character, especially if you know a canon characters appears in the chapter of where you took the unique phrase from. In doing this, you're narrowing down your search and catching anyone who stole it. You don't know if they copy it word-for-word or switch words around but doing it both ways helps to cover both possibilities.
Google isn't the only way to search for plagiarist there are other tools such as Copyscape. Copyspace searches through Google much more efficiently because all you do is punch in the URL of the work you want to check and click submit. Plagium is similar to Copyspace, doing the same thing and it's free, giving you unlimited results. But keep in mind that it uses Yahoo search engine which will produce different results then Copyspace or Google.
7. What to do if you found someone who plagiarized your story?
First things first, it helps to print screen what they plagiarized from you. That way you have proof to show off that they did take something. And it helps to use it to compare to the original (what you wrote) to prove to them and to their “fans” that you are not lying. Don't forget to make sure you screecaps the dates of yours and “theirs stories” of when they were published. After all, the dates help to show people that yours is the original and has been out longer then “theirs”. Plus you never know if they'll quickly delete it and/or try to blame you for hacking them or something ridiculous. It's better to be safe then sorry by having proof and being ready incase that happens.
Then the next part is the confrontation which can be easy or difficult depending on the plagiarist. Some will ignore your message and do nothing, some might ignore the message and delete everything or they might respond back telling you off and claiming you're wrong or something.
Either way, just be sure that you contact them. Tell them who you are, why you're contacting them and calmly educate them that what they did is wrong. Be stern as well because you don't want to sound like someone who could easily be push around. It's important to try to reach them first and see if you could politely tell them to delete or change whatever they copied from you. If they refuse to respond or are being extremely difficult or they won't take any action then you need to contact the website owner, (or a mod, or a administrator, or a manager,) or other authority on the website. If you discuss with them what has happen they will take the proper action in handing the plagiarist. And they WILL take action because no site wants to be known for allowing plagiarized content. It denigrates the reputation of the site and among other reasons as well.
Also, if you're not the writer of a fan fiction but came across a story that you know has been plagiarized you do pretty much the same. The difference is that you first contact the original writer of the fan fiction and tell them about it. Let them know where it is, how you noticed it was copied and that you wanted to contact them first before reporting taking action with the plagiarist (because if you contact the plagiarist first they could quickly erase their evidence or change it completely before you could tell the original fan fiction writer about it). So it's best to let the writer know so they can screen caps/gather evidences (or you can do it for the author before contacting the plagiarist or reporting them to the website authority) incase the plagiarist or the website owner are being difficult and need proof.
8. Ways to prevent people from plagiarizing your stories?
There's no fool-proof plan or way to prevent anyone from stealing your stories. But the best way one can try is to make it known on your profile where your stories are published at and what username you use. That way if someone stumbles upon your story somewhere else that you didn't list and it seems fishy then they can check your profile to make sure. So if they are seen anywhere else that you didn't list then they know you were plagiarized.
Another way some people do is go to all the major fan fiction archive websites, create an account and post their stories. They feel that by doing so it would help to expose their stories so that if someone else sees the same story (or nearly the same) in another name on the same site as well, then they know something is wrong. Otherwise if the author didn't post in that site then the plagiarist would post there because it would increase their chances of not getting caught. After all, many would just assume that it's the same author with a different name.
9. Plagiarism or Coincidence?
There are many cliches in fan fiction and many come across stories that contain those cliches. But remember, they aren't plagiarized because cliches are common knowledge. Plus, don't be surprise if a couple of stories contain similar ideas! Anyone can think of a broad concept but it's in the details that we can find plagiarism.
If you came across a story that was about two female twins switching places/lives and trying to bring together their lives/families then don't be alarmed if you see another story with a similar concept but with different details. Like if there's another story about twins only they are a male and female. They must hide their real gender and disguise as the other when they switch lives.
You see, the concepts are similar but it's the details that let you know if someone copied or not. If it is obvious they took your ideas and copied your characters, situations and plot lines then you might have a case of plagiarism and not a coincidence.
 — The example used for paraphrasing belongs to: HERE
If you want to learn more clink: HERE