When you encounter the police, you do not want to incriminate yourself in any way. You can do this by accidentally saying something that makes you appear guilty -- whether you are or not. One little statement can lead to an arrest and may land you in jail. This is not the time for any mistakes.
The advice you get from a lot of people is to "plead the fifth." This is in reference to the Fifth Amendment, which says that you have a right to remain silent. You don't have to say anything, specifically so that you do not feel forced to incriminate yourself. You can stay silent.
However, some legal experts -- such as one law professor -- say that even staying silent isn't in your best interests. You don't want to talk, but you should just say one thing:
"I want my lawyer."
The police know that you have a legal right to representation. You can tell them that you will not give any statements or answer any questions without your lawyer present. You're not going to waive that right and say something in the heat of the moment that may come back to haunt you.
The police may not like this, as they want you to talk, but you're not really being confrontational. You're not even saying that you won't talk or you won't answer questions eventually. You're just saying that this is not the setting in which you want to do it. You know that having a legal team on your side can help you answer properly, and you're not going to risk it just to talk to the police a little sooner.
After all, the police have a lot of tactics that they use to get people to say the wrong thing. They may try to intimidate you. They may threaten you, acting as though talking is the only way to get yourself out of the situation. They may also lie to you and attempt to trick you into saying something you never would have said otherwise.
During an encounter or after an arrest, you're at a bit of a disadvantage. You may feel like your best bet is just to cooperate and see if they'll let you go. However, doing that could be the exact opposite of what you should really do, and this is really not the time for mistakes.
When facing serious accusations or charges, make sure you know what legal options you have. Make sure you understand your rights. Make sure you know what you should not say and what you should say to the police.