The warning given by American policemen to suspects being arrested is called a Miranda warning, because its use was mandated by the 1966 United States Supreme Court decision in the case of Miranda vs. Arizona.
In 1963, Ernesto Miranda was arrested for robbery, kidnapping, and rape. He was interrogated by police and confessed. At trial, prosecutors offered only his confession as evidence and he was convicted. The Supreme Court ruled that Mr. Miranda was intimidated by the interrogation and did not understand that he had the right not to incriminate himself and the right to have counsel, and they overturned his conviction. Miranda was retried, and this time the police did not use the confession but called witnesses and used other evidence. Miranda was convicted, and served 11 years.
Quote from the ruling: "...The person in custody must, prior to interrogation, be clearly informed that he has the right to remain silent, and that anything he says will be used against him in court; he must be clearly informed that he has the right to consult with a lawyer and to have the lawyer with him during interrogation, and that, if he is indigent, a lawyer will be appointed to represent him."
This principle of law, though under different names, has been adopted in jurisdictions throughout the English speaking world.
As a result, we have a new verb in the American vocabulary, "to mirandize" meaning to read to a suspect, held in custody, his Miranda rights.