Many people believe that if they are arrested and not "read their rights," they can escape punishment. Not true. But if the police fail to read a suspect his or her Miranda rights, the prosecutor can't use for most purposes anything the suspect says as evidence against the suspect at trial. Of course, as with nearly all legal rules, there are exceptions. (See When Police Violate the Miranda Rule and Exceptions to the Miranda Rule.)
The Miranda warning (from the U.S. Supreme Court's Miranda v. Arizona decision), requires that officers let you know of certain facts after your arrest, before questioning you. An officer who is going to interrogate you must convey to you that:
It doesn't matter whether an interrogation occurs in a jail, at the scene of a crime, on a busy downtown street, or the middle of an open field: If a person is in custody (deprived of his or her freedom of action in any significant way), the police must read the Miranda rights if they want to ask questions and use the answers as evidence at trial.
If someone is not in police custody, however, no Miranda warning is required and anything the person says can be used at trial. Police officers often avoid arresting people—and make it clear to them that they're free to go—precisely so they don't have to give the Miranda warning. Then they can arrest the suspect after getting the incriminating statement they wanted all along.
For more detail on these issues, see Miranda: The Meaning of 'Custodial Interrogation.'
Do you have to respond to police questions if you haven't been arrested? Generally, no. (You typically don't have to answer even if you're under arrest.) A police officer generally cannot arrest a person simply for failure to respond to questions. (There are, however, situations where you might have to provide information like identification; see Daniele Watts Controversy: Can Officers Demand ID?)
The almost-universal advice of defense attorneys is to keep the old mouth tightly shut when being questioned after an arrest, at least until after consulting an attorney. Suspects all too frequently unwittingly reveal information that can later be used as evidence of their guilt.
Without a Miranda warning, what the arrestee says in response to custodial questioning can't be used for most purposes as evidence at trial. For more detail, see Statements Obtained When Police Violate Miranda: Can They Be Used Against You?
A violation of Miranda rights doesn't necessarily mean that the officers coerced the statement out of the suspect. But if they did, not only is the statement inadmissible, but so too is any evidence that the police obtain as the result of it. (To learn more about coerced statements, see Miranda and Involuntary Confessions.)
The Miranda rule is complex, and no one article can address all its ins and outs. If you've been arrested or charged with a crime, you should talk to a lawyer for a full explanation of the law, including how it may differ slightly in your state.