The way I understand these things... and Jason will correct me if I am wrong... is that the remember me checkbox is saved in a cookie. If you have cookies disabled, or you clear your cookies through Tools Options menu (Firefox, IE is slightly diff) that clears the cookie and the site won't remember you.
If the cookie remains and you checked that button, you should be logged back on automatically for a certain period of time (the cookie has an expiration date associtiated with it). Once the cookie epxires (seems to be a week or so with mtbnj) you have to re-log in.
If you're logging out, all bets are off above... you've told the site ah, I changed my mind forget about remembering me.
This isn't your e-bank site, don't logout.