Most people know that they have the right to remain silent during an interrogation by the police. It's guaranteed by the Fifth Amendment to the Constitution.
Unfortunately, not many people understand that they have to invoke that right clearly and completely in order to really be protected.
Here's what often happens instead: A police officer reads someone the Miranda warning that includes the reminder that they have the right to remain silent. The officer then says, "Do you want to keep talking with me?" The suspect then says, "I'm not sure. What do you think?" The police will naturally take advantage of this situation every time!
Alternately, someone might start out strong by saying, "I don't want to speak with you without my lawyer." Then, a few moments later, doubt will creep into that person's mind. Some people think that it might be possible to clear up the whole issue if they just go ahead and talk with police for a little while and explain their innocence -- so they start talking again and end up giving the police information that is used against them.
You never want to be in the position where a judge has to decide if you really invoked your right to remain silent or not -- or whether you ultimately rescinded when you started talking again. The only clear way to invoke your rights is to make a clear, simple statement like, "I don't wish to talk to you. I would like to speak to my attorney."
At that point, stop talking -- even if the police keep talking to you! Keep in mind that the police know that it's very uncomfortable for most people to refuse to talk to them. Even in such an adversarial situation, it feels rude and unpleasant. The police also know that people often wonder if they're making a mistake by staying silent. Expect them to try to play on both of those feelings.
No matter how uncomfortable it makes you feel, stay resolved and stay silent. You don't want to create any sense of ambiguity about your intention to invoke the Fifth Amendment. It's much easier to mount a solid defense against criminal charges when you've done your part to protect your rights.