—Tom Lehrer, "Lobachevsky"
Plagiarism is essentially taking the work of others and attempting to pass it off as one's own. In academia, it is generally defined as putting forth ideas cribbed from other places and claiming that the writer came up with them -- in other words, discussing facts and theories without proper citation. This can get dicey if the author of a paper or essay happened to come to the same conclusion as a previous writer independently. It can also lead to "Plagiarism Paranoia", when a student panics over whether they've cited it enough or not. However, around here, we're more concerned with literary plagiarism. For plagiarism used as a plot point, see Plagiarism in Fiction.
When it comes to works of fiction, the term "plagiarism" tends to be misused quite a bit; indeed, most of what is referred to as "blatant plagiarism" is actually far less blatant than actual plagiarism. If two stories happen to have similar elements, it does not mean that one plagiarised the other. All stories are influenced by what has come before and what the author has experienced; the most likely case is that one story inspired at least part of the other.
Literary plagiarism can involve the use of events and characters from a work of fiction, or the wholesale copying of another author's text. To be plagiarized, the text, characters or ideas must be used without crediting the original author for their work.
Note that plagiarism does not necessarily involve Copyright infringement, or vice versa. A writer can plagiarize a work that is no longer under copyright or was never copyrighted. If writer Joe Smith publishes a play he copied from a William Shakespeare anthology, Smith is still plagiarizing even though Shakespeare's work is in the public domain. Copyright infringement occurs when Joe Smith uses someone else's copyrighted work without their permission, even if Joe Smith is upfront about the fact that he is not the author.
Do not confuse Trademark infringement with Copyright infringement. While almost all Fan Fiction of works not in the Public Domain involves (of necessity) trademark infringement, it would be safe to say that very little fan fiction involves plagiarism (and thus copyright infringement) of original sources; it makes no sense when one's target audience would recognize Canon material immediately. What little copying exists is typically on the level of Fair Use. (It is not unheard of, though, for would-be fanfic authors to plagiarize works from other fanfic authors.)
It is sometimes debatable if a similar plot, introduced in a different setting and written differently with most of the details changed, still counts as plagiarism. The Inheritance Cycle is an example of this, with the first two books having a plot line that is quite similar to Star Wars but with many different details and an entirely different setting. Whether or not Christopher Paolini is actually guilty of plagiarism is hotly debated between hardcore fans and Fan Hater.
On the other hand, a story about a farmboy who rescues a princess and destroys the Big Bad's Doomsday Device is not necessarily plagiarism; if it's different enough from Star Wars, all you can say is that they both make use of some of the same tropes (and, indeed, George Lucas cribbed from quite a few older sources).
A Shout-Out is probably the least controversial. In most cases, a shout out only lasts a second or so, and it is clear from the context that the person who wrote it did so mainly to acknowledge the influence or awesomeness of one who came before.
A Homage is more difficult to defend. In general, a homage will use imagery and ideas from the original material but with sufficient differences that one can easily tell that the new work is influenced by what has gone before.
For example, the fifth in Stephen King's Dark Tower series sees the heroes defending a town against raiders who attack every so often, in a plot which was acknowledged in the series itself as being inspired by both The Seven Samurai and The Magnificent Seven. However, the characters were original, and the setting and raiders were largely the product of King's own imagination (albeit containing numerous shout outs).
A "remake" or "retelling" is doing the original work over again with some relatively minor changes, while openly admitting that it's heavily derived from the original. A famous example of a retelling is John Sturges' film The Magnificent Seven, which had the plot and even some of the same dialogue as Akira Kurosawa's earlier Seven Samurai. The only real difference is that while Kurosawa's film was set in feudal Japan, Sturges' was set in The Wild West. Why is this not plagiarism? Because Sturges had two things going for him: Kurosawa knew what he was doing and gave his approval, and Sturges openly acknowledged that he was simply moving Seven Samurai to the Old West. He never claimed that it was a purely original work.
Contrast this with Sergio Leone's A Fistful of Dollars, which is a frame for frame remake of Kurosawa's Yojimbo, but set in Spain pretending to be Texas. When Kurosawa saw the film, he wrote to Leone "It is a very fine film, but it is my film." He sued and won, because Leone didn't have permission, didn't give credit, and didn't pay.
Finally, a parody is the hardest thing to guard against accusations of plagiarism, because parodies tend deliberately imitate the thing they're making fun of in order to get jokes. In many cases, particularly on the Internet, accusations of plagiarism are avoided simply by sticking a note saying something along the lines of "This is a parody. [work being parodied] is the intellectual property of [copyright holder]" at the top of the story or sidebar of the YouTube video. This is used by Yu-Gi-Oh the Abridged Series and its imitators. The doctrine that is invoked to protect parodies (and other forms of copying) is called Fair Use.
If a book parodies another work of fiction, there is generally an introduction in which the author clearly states that this is a parody, and explains why they are making fun of the original work. This approach is used in Bored of the Rings, a parody of The Lord of the Rings, among others.
Parodies in TV and movies often eschew both of these in favour of simply changing the plot just enough so that it technically counts as an independent work. In other cases, the work is blatantly a parody of something else, and the authors generally argue that parodies are protected under the Fair Use doctrine.
- Use of the word "cribbed" comes from the fact that plagiarism literally means baby stealing.