Historically, suspects taken into police custody didn't need to use any magic words to let police officers know that they wanted to exercise the right to remain silent. It was probably enough to simply say nothing in response to police questions. But in June 2010 the Supreme Court ruled that a suspect who stays silent and uncooperative after Miranda warnings are given has not invoked the right to remain silent. Some kind of simple but unambiguous statement is the key, the Court said in Berghuis v. Thompkins. In other words. you can effectively give up your Miranda rights by simply failing to invoke them while the police continue to question you.
So, after an officer gives you a Miranda warning, it's best to make your desire to remain silent clear (and hopefully stop any police questioning) by saying something like "I wish to invoke my right to remain silent."
Read the full text of the Supreme Courts decision: Berghuis v. Thompkins