As most people know, the 1966 case of Miranda v Arizona held that the police are required to notify a suspect who is in custody that he has the right to stay silent and that if he speaks, any statements can and will be used against him in a court of law.
But of course if that is all the court had said, the Miranda decision wouldn't mean a whole lot. Just saying someone has a right to not make a statement would be meaningless sophistry, a right without a remedy, unless there are sanctions to back it up. The significance of the Miranda decision was not just that people in custody are required be notified of their Fifth Amendment right against self-incrimination, but that if that right is violated their statements cannot be used against them at trial.
This second part of Miranda was an extension of a series of decisions stretching back decades earlier which held that evidence that was obtained in violation of a person's Constitutional rights could not be used at trial. This was definitively articulated by Weeks v US for Federal cases and by Mapp v Ohio for state cases.
This doctrine is known as the exclusionary rule. It is one of the cornerstones of American jurisprudence and has been wildly controversial. The underlying premise of the rule is that unless law enforcement officials are severely penalized when they violate a suspect's Constitutional rights, they have no incentive not to keep a violating them. Excluding unlawfully gained evidence from trial is viewed as the best tools to make law enforcement officials abide by the rules since both from the standpoint of the police as well as the prosecutors all of their work will come to naught if they are unable to get a conviction. The courts did analyze other potential remedies, such as sanctions against police officers, but determined that they are largely unworkable, and provide no real protection to a defendant whose rights have been violated.
It is important to note that the Fifth Amendment right to not be "compelled in any criminal case to be a witness against himself" exists regardless of whether or not the Miranda warning is given. If the suspect tells the authorities that he does not want to make a statement or in fact does not make a statement, his Fifth Amendment right against self-incrimination protects him. All Miranda does is ensure that the suspect is aware of that right.
Miranda applies when a suspect is in custody because the courts have determined that the very fact that a person is in custody is inherently coercive. If a person in custody freely and openly decides to talk after being told that he does not need to talk then the coercive element is taken away.
There is one other factor about the Miranda rule that is critical. Miranda does not prevent the authorities from questioning a suspect who has not been told of his Miranda rights. It merely prevents the authorities from using any of the statements made by the suspect and certain evidence that flows from those statements which is often referred to as the "fruit of the poisonous tree" from being used against the defendant at trial.
Now, what about the public safety exception. In a 1984 case, New York v Quarles, the Supreme Court set forth what is referred to as the "public safety" exception to Miranda. The court referred to it as a "narrow" exception which would allow the police to question a suspect before he is given his Miranda warnings about limited matters directly affecting the safety of the officers or the public. Further, the court held that any statements made by the defendant in response to those could be used at trial.
Quarles was a 6 to 3 decision with a vigorous dissent. The dissenters questioned whether or not, on a purely factual basis, there was a public safety issue in the case But more importantly were very concerned about how this exception could be abused. They asserted that this exception would cloud the clear unambiguous requirements of the Miranda rule which they believed had served the public and the interests of justice well for many years. In response, the majority stated
the exception will not be difficult for police officers to apply because in each case it will be circumscribed by the exigency which justifies it. Police officers can and will distinguish almost instinctively between questions necessary to secure their own safety or the safety of the public and questions designed solely to elicit testimonial evidence from a suspect.
There is an important point to remember when analyzing the public safety exception. It involves both factual and legal questions. If one does not believe that a public safety exception should exist, then no set of facts will justify such an exception. On the other hand if one's concerns are purely fact related, i.e., whether or not there a clear and immediate public safety concern justifying the questions, those concerns can only be resolved by a clear understanding of all of the underlying circumstances.
The public safety exception has been in existence for almost three decades. Needless to say the FBI and the Justice Department had a keen interest in providing guidance to its agent about the exception since they do not want to have prosecutions thrown out based on improper questioning. To that end, the FBI has issued a fairly comprehensive explanation of public safety exception which is a useful primer on how the exception developed, and how the FBI views it in application.
In addition, the FBI has issued specific public safety-terrorist guidance that is designed to be used in terrorist cases. Here is the meat of the guidance contained in that document, which is titled "Custodial Interrogation for Public Safety and Intelligence-Gathering Purposes of Operational Terrorists Inside the United States."
1. If applicable, agents should ask any and all questions that are reasonably prompted by an immediate concern for the safety of the public or the arresting agents without advising the arrestee of his Miranda rights. 
2. After all applicable public safety questions have been exhausted, agents should advise the arrestee of his Miranda rights and seek a waiver of those rights before any further interrogation occurs, absent exceptional circumstances described below.
3. There may be exceptional cases in which, although all relevant public safety questions have been asked, agents nonetheless conclude that continued unwarned interrogation is necessary to collect valuable and timely intelligence not related to any immediate threat, and that the government's interest in obtaining this intelligence outweighs the disadvantages of proceeding with unwarned interrogation.  In these instances, agents should seek SAC approval to proceed with unwarned interrogation after the public safety questioning is concluded. Whenever feasible, the SAC will consult with FBI-HQ (including OGC) and Department of Justice attorneys before granting approval. Presentment of an arrestee may not be delayed simply to continue the interrogation, unless the defendant has timely waived prompt presentment.
What does all this mean in the upcoming interrogation of Dzhokhar Tsarnaev, the one surviving brother accused of carrying out the Boston Marathon bombing. It is possible that the FBI has some legitimate basis to invoke the public safety exception based on concerns about whether Tsarnaev has planted other explosive devices or is aware of other specific pending terrorist attacks. Beyond that, it's difficult to imagine any questions that would fall within the exception.
That does not mean, of course, that they can't ask him about anything they want to without having previously advised him of his Miranda rights. They can do so with the full awareness that in all likelihood they will be unable to use any statements he made about those matters in any trial against him.
Also, if he advises the authorities that he does not want to speak to them, or simply does not speak to them, even though they have not advised him of his Miranda rights, he is perfectly entitled to do so and the authorities are prohibited from compelling testimony.
We should always remember that even though the FBI may think it's questions are reasonable and fall within the public safety exception that does not mean that the courts will agree with them. Also, any answers to questions they may ask about future threats, whether admissible or not, don't necessarily in any way impact the admissibility of all of the evidence that has already been gathered in the case or that make be developed in the future that is unrelated to any statements.
Given some comments I've read I need to make some editorial observations. This is not Guantánamo. This is not some black site in Afghanistan. The United States has successfully prosecuted a large number of people accused of terrorism in the Federal courts of the United States and has done so while fully complying with the Constitutional guarantees applicable in such cases. Lindsey Graham notwithstanding, this case will be treated just like all of those other cases.