Let's say I find an article online. I contact the writer of this article and, without expecting anything in return, they agree to give me full, unrestricted rights to use any portion of their article, including directly using quotes from the article, without citing it. The agreement even allows me to claim the work as my own, even though it isn't.
Because I have all the rights to do this, would this be considered plagiarism?
It doesn't really matter that the author gave you "unrestricted rights to use any portion of their article" without the need of citing it. We, the readers, want to know that you're using it.
Because I have all the rights to do this, would this be considered plagiarism?
Yes.
Yes, it is plagiarism. The permission is irrelevant.
Consider this simplified but actually equivalent example: if your neighbor allows you to copy her answers in a written exam, is that not cheating just because she allowed you to use her answers? Of course not, that is ridiculous! It is cheating simply because you didn't come up with the answers yourself; whether or not you had your neighbor's permission is completely irrelevant.
You are confusing two orthogonal things: plagiarism and copyright violation.
Plagiarism is an ethical concept; it refers to passing off other's work as your own. Copyright violation is a legal concept; it means using a creative work in a way that is reserved for the copyright holder without acquiring a license from the copyright holder.
Those concepts are orthogonal: if you copy large portions from a book and you properly cite and attribute them, it's still a copyright violation even though it's not plagiarism. And in your example, it's still plagiarism even though it's not a copyright violation.
Note: plagiarism may come up in a legal context as well. For example, in most institutions of higher education, there are rules forbidding plagiarism, if you are a student, you often have to sign or acknowledge an honor code forbidding plagiarism, if you are faculty, a tutor, a TA, an employee, there might be a statute about plagiarism in your employment contract, making it a finable or fireable offense. Also, your thesis will have a signed statement that you did not plagiarize, and if you did, signing that statement may be considered fraudulent. In that case, plagiarism may not only have ethical/moral dimensions but a legal one as well (you get expelled / fired / fined / your degree revoked). In the case that the degree is a prerequisite for practicing in a legally protected profession (e.g. lawyer, doctor), falsely signing that document may even be considered a criminal offense or even a felony.
Plagiarism is passing of the work of someone else as your own. So regardless of the authors permission, this is absolutely plagiarism.
This is quite similar an increasingly common form of plagiarism where students purchase an essay or pay someone to complete their assignment for them. At the universities I've been involved in this is treated very seriously and students fail entire units when they are caught. As serious academic misconduct they can in fact be removed from a course entirely.
Even if you yourself are that other author, you still need to cite your old work in your new.
Citation has nothing to do with copyright. You cite your sources to give attribution to the ones before you, to let your readers know which giants' shoulders you stand upon and to differentiate the parts you wrote from the previous works of others, avoiding passing their work off as your own. It's perfectly fine to cite something that nobody in the world is allowed to read due to copyright reasons or what have you (allthough a bit mean to your readers).
Plagiarism is committed when you don't cite the article regardless of any written consents.
Copyright is violated when you reproduce portions of a published article in the sense described by a copyright law.
Stealing intellectual property may be committed when you reuse ideas from the article without the consent of the owner of the intellectual contents of the article.
The writer cannot protect you from 1. In usual cases, from 2 neither. In certain cases, his/her consent might be your protection from 3.
Said that, don't copy-and-paste. Rewrite, quote, and cite properly.
Others have pointed out that something can be plagiarism without violating copyright.
In addition, even if an author gave you permission to use something from an article it may well still be a copyright violation because in many cases when you publish in a journal you also sign over the copyright. As such even if the author gives you permission, they may not be the copyright holders so legally it's not theirs to give.
[Other answers have pointed it out - this answer attempts instead to put it in terms that someone who simply doesn't understand the difference between copyright violation and plagiarism could understand]
This happens every single time a work is placed in the public domain.
Every work by or for the US government; every work produced before some dates and some works thereafter; every work explicitly licensed CC0.
All of these works grant complete ability to reproduce, remix, and reuse the entire work without credit. Doing so is perfectly acceptable.
Whether it is OK to do this, however...
Consider that any one of us may reprint the entire works of Shakespeare and put our own name to them, without crediting Shakespeare. Or Nabokov. Or translate the Illiad or Odyssey to English without crediting Homer. Or...
And then consider the fact that despite it being perfectly 100% legal, nobody ever does this.
In the field of academia, plagiarism is seen as a far greater sin than elsewhere, so the backlash would be far worse.
[Edit: You could argue that you were using only some small, fair-use portion of a document in a copyright sense. But that could still get you an accusation of plagiarism in adademia.]
One general comment about plagiarism:
If an African-American says, "I have a dream," any allegation of having plagiarized MLK is ludicrous. Such allusions are shorthand quotations.
Plagiarism is presenting someone else's work as your own, according to the combination of stated and implied claim to originality, without explicitly marking the quote. An allusion is not intended to be perceived as original; it is intended to be a shorthand quote, and it just does so without quotation marks as a metaphor drops the word "like" from a simile so that "time is like a river" becomes "time is a river".
I found it rather ludicrous when Orson Scott Card was accused of plagiarism in incorporating bits of story that are apparently well-known to people who know Mormon Scripture. I am not a Mormon and personally have no interest in reading Mormon Scripture, but there seemed something extraordinarily confused about what, to Mormons or non-Mormons who understand that tradition, is a common theme. Neither my own The Sign of the Grail nor any other work since the twelfth century Historia Regum Britanniae or the Brut pseudo-history is simply guilty of plagiarism by vice of using the character of Merlin without explaining that Merlin was taken from earlier works (and the Brut is probably not original; Merlin's name was altered from an earlier Celtic character name apparently to avoid collision with a French swear word).
(And as those of us in the U.S. are nearing an election, I might point out that it is a thoroughly established practice for anyone remotely near POTUS candidacy to use speechwriters, and needless to say I have not heard anyone indict anyone else for giving a speech written by a speechwriter without an academic style of attribution. On a much less influential scale, authorship is a bit fuzzy in the business world; there is such a thing as unacceptably stealing credit, but most of a company's messages to its audience do not include attribution of who made the copy, who made the graphics, who did XYZ in the background.)
The standards for plagiarism are different in academia from other places; for me it has been only in academic settings that people have been concerned about what constitutes plagiarism. I've run into people who care about honesty and I've run into people who thoroughly believe in giving credit where credit is due (one CEO asked me to co-sign a letter to customers about a new offering, although that is a novelty). And this question isn't really in a grey area; it arises from failing to distinguish plagiarism from copyright issues. But I have only seen fine-grained discussions of shades of grey about plagiarism in academic circles, where a good probable rule of thumb is probably, "If you can express quotation or debt to a source you have used, however unimportant, footnote it."
Regrading the authorship:
If you buy Pablo Piccasso's paintings and copyrights to them, you own the paintings, you can earn money for displaying them etc., but they must be creditted as Pablo's paintings, not yours. Period.
Regarding the copyrights:
When a paper is published the copyright to the paper usually belongs to the publisher. Copyright to property, that the article is based on, belongs to the writers.
In your case you are entitled to use, edit and make profit based on their work freely. Still, they shall be creditted for their work.
Approaching from different point of view:
Scientists are said to be people with higer knowledge, wisdom and morality than the average folks. At least, not citing actual authors is immoral.
My two cents: If you have to ask if something is plagiarism, consider it plagiarism.
Always err on the side of caution. An ethics/honor code violation is not something you want.
If the "author wants or needs to not be named," I would cite this way (Anon. 2016). The format is my school's style guide; the content is my application. In USA, "fair use" limits copyright slightly, but I ould still consider an author's wishes if I thought there were a chance he/she might be in a situation where they must protect their identity or their location.
I would never pretend someone else's words were my own. Although, I must admit there were times I have read something after saying it myself and wondered whether I had come up with it independently or whether I had read the work earlier and forgotten.
Anonymous (Anon.) 2016. Private communication (author wishes to remain anonymous).
While I generally agree this would be plagiarism, in some circumstances it would not - and not only in the allusion case (see @JonathanHayward 's answer).
Specifically, if you believe and have the author's concurrence that part of the article is folklore, or should not be considered an integral part of the article somehow and not really attributed to the author, then it's possible you're not committing plagiarism.
But you really should:
Note also that you can make a qualified citation, e.g.
\footnote{This is discussed in Prof's John Smith's monograph on
the Frobincation of Bars \cite{JSFrob}, but blah blah blah}.
