Remember Me doesn't do anything with passwords. What it does is drop a cookie so you can come back to the site without logging in. If you've clicked remember me, you shouldn't even ben asked for your password on subsequent visits (provided nothing clears the cookies). It will just say "Welcome Warren" where you would normally see the login fields.
If you see a user name and/or password inserted in our login fields, it is because your browser is remembering it and inserting it there.
If the remember me (as I described it) is not working, it's possible you are blocking cookies.