quotation out of context
The fallacy of quotation out of context (aka contextomy) occurs when some argumentative force of a quotation is achieved by leaving out the context in which the quotation occurred. The omitted context could be the preceding or succeeding words out of which the quote was extracted, or the situation in which the quote was originally stated, or even the tone of voice of the quote when uttered, e.g. sarcasm or interrogative (question-asking) intonation.
Here is an example:
Movie review in Film Craze Magazine: The only thing surprising and new about the idiotic plot is that it's even worse than that of the director's last movie.
On a poster promoting the same movie: "...surprising and new..." -- Film Craze Magazine
The problem is that the quote on the poster reads like unqualified praise, when it is actually part of a negative criticism.
When an arguer commits this fallacy several times in succession to create the impression that a broad variety of sources are saying a similar thing (when in fact they are not), it is often referred to as "quote mining."
Note that when the item that is taken out-of-context is a non-verbal action or event, the more general name "contextomy" can be used for this fallacy.
An example of that would be:
Letty: Jane must be a violent person, because there's this video of her punching a clown at the circus. See? I have it right here!
Betty: Wow. My feed has a longer version of that video, where you can see how the clown puts his hand up her skirt right before she hits him.
Letty: Oooohhhhhh.