What is the name of these well-known lines?
You have the right to remain silent. Anything you say can and will be used against you in a court of law. You have the right to an attorney and for an attorney to be present during questioning. If you cannot afford an attorney, one will be appointed for you.
I think this is the Miranda-Escobedo caution named after two American court cases where these things were not made clear to the arrestee. Also now known (at least on TV cop shows) as "mirandising" the suspect
Woofgang is correct. They are commonly referred to as the "Miranda rights", and are read to a suspect as soon as he or she is taken into custody and before he is questioned. The original case was actually Miranda versus the State of Arizona.