Also known as the "Miranda Rule, this is the name given to the requirement that police officers, in the U.S.A., must warn suspects upon arrest that they have the right to remain silent, that any statement that they make could be used against them in a court of law, that they have the right to contact a lawyer and that if they cannot afford a lawyer, that one will be provided before any questioning, if so requested.
Failure to issue the Miranda warning results in the information or confessions and the like, so obtained, to not be admissible as evidence in the court.
The warning became a national police requirement when ordered by the US Supreme Court in the 1966 case Ernest Miranda v. Arizona 384 US 436 (1966) and that is how it got the name.
The US Supreme Court has an excellent web page on it at oyez.org/cases/1960-1969/1965/1965_759/, which includes a link to the actual decision, as well as this summary:
"The Court held that prosecutors could not use statements stemming from custodial interrogation of defendants unless they demonstrated the use of procedural safeguards "effective to secure the privilege against self-incrimination." The Court noted that "the modern practice of in-custody interrogation is psychologically rather than physically oriented" and that "the blood of the accused is not the only hallmark of an unconstitutional inquisition." The Court specifically outlined the necessary aspects of police warnings to suspects, including warnings of the right to remain silent and the right to have counsel present during interrogations."