"Remember Me" is typically used to remember the email address/user name of the user that logs in, so they only need to enter their password on subsequent visits. Contrary to other answers, I would say that the password is almost certainly not remembered as a matter of practice. Alternative forms of this are "remember my email" or "remember user name", which makes it clear that they will need to enter their password again. This function is usually (more or less) safe on a public device; while it shows the user name, the person would need to also know the password to access a given account.
"Stay Signed In" means just that: the user's session token is preserved with a very long expiration time and won't automatically be cleared out when the user closes their browser. This function is never safe on a public computer, and it should be made clear to the user that they're logged in indefinitely when they choose this option. It is less secure than a simple "remember me" setting, but offers convenience for users logging in from private devices.