This can be called, pleading the Fifth, taking the Fifth, or invoking the Fifth. It may also be called invoking you right to remain silent. The witness can use any of these phrases, or they can say, "I refuse to answer on the grounds that it may incriminate me." If the witness is worried about public perception, they may also insert the phrase, "On the advice of my lawyer..."
The Fifth Amendment of the U.S. Constitution states that no person "shall be compelled in any criminal case to be a witness against himself."
What this means is that the government has to build a case against you, using evidence that does not come from your mouth. Of course, you can waive this right and talk to the police, and give them a bunch of evidence to use against you if you want. And many, many people do exactly that. But you have a constitutional right not to do so if you don't want to.
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