"Remember Me on This Computer ?"
/Your computer can remember a lot about you. And that can make it easier to log in to websites.
Sometimes that is OK and sometimes it might send information you don’t want to send. Or, if it is a shared computer - or you want to protect your identity if the computer is stolen - you may want to re-enter the log in information, every time. It can be a pain but, critical sites like banking, will keep you better protected.