The remember me box deals with saving your password for the website, so if you already had it saved prior you might not get it. If you are having problems with having to login to vbulletin whenever you access a board even if you are already logged in on the homepage, however, I don't think the remember me box will fix it. I have had the same problem on 3 different computers with linux/firefox, windows 7/firefox, and vista/firefox.
Also just a side note, but firefox is up to version 9.0.1. It goes pretty fast now. You might want to update for security reasons as I doubt 3.65 is still developed.
It's only a vBulletin login option. There are several places you can log into the site -- in the header (no remember-me option), when accessing a pay article (no remember-me option), and via vBulletin (remember-me option should be included).