For example, you see someone in front of you hitting someone else ….. and call the police.

This is perfectly OK in most cases BUT it would be being taken out of context if the two fighters were in a boxing ring.

Or it can refer to quoting some sort of literature, when you use a small portion of something to try to define the author’s intent or the meaning of it without understanding the context, what is said around it.

for example, ‘quoting’ someone as saying “Abortions are lawful”, when their full sentence was “The claim that abortions are lawful is still to be proven”

…or it could mean interpreting words without regard for the situation and circumstances under which they were said…

for example, quoting a professor stating that native americans were violent savages, without including the fact that he was explaining the 16th century european view.

So to “take things out of context” means to take a word or phrase and interpret it, apply it, or even twist it to have a different meaning than the original spoken or written intent.

1. The example of the Bible.

The other answer provided an excellent example. You cannot take the Old and New Testament “out of context” but should interpret these together in historical context — the Old Testament depicts the history of war, corruption, genocide and death by living by the “letter of the law” and “retributive justice” (so it is a warning NOT to live that way); while the New Testament presents the difference in living by the “spirit of the law (truth and love)” and “restorative justice” (instead of unforgiveness, hatred or revenge). So the point is not to be stuck in the old ways, but to see why it causes suffering to live under material greed, and to seek liberation from that.

a. “eye for an eye, tooth for a tooth” — by the old ways of justice, people used to wipe out a whole family as revenge for killing one person. So the original meaning of this law was to “limit” the compensation to “one-for-one,” to be fair. So this law was originally intended to avoid excess revenge, and to make things more just.

b. “turn the other cheek” — in the old days, slapping someone on the left cheek implied a backhanded slap for a slave or noncitizen. So the meaning of asking to be slapped on the “right cheek” is to insist that if someone is going to correct or rebuke you, at least address you as an “equal” as a citizen or peer. So this does not mean act as a “doormat” and let people walk on you. It means to bring humility by respecting people as equals, that we rebuke and correct each other on equal ground, for the purpose of reconciliation or “restorative” justice, not for revenge or “retributive justice” and not for tolerating abuse either.

[The other answer makes this point, but did not explain the full history behind these two quotations used from the Bible.]

2. A more recent example (concerning 9/11):

One man who heard the airplane that hit the Pentagon stated clearly it was an airplane but it “sounded like a missile.”

This quotation was taken “out of context” where it is abused in propaganda as “proof” that it was NOT an airplane (as the witness actually testified that it was) but was a missile!

I understand this man is “devastated” that his words have been taken out of context to mean the opposite of what he witnessed. He said it was an airplane, but because he metaphorically compared the frightening sound to a missile, now people are twisting his words to spread conspiracy theories that a missile hit the Pentagon.