So on most of the websites that require login information that I go to, I have "Remember me" or the equivalent turned on. Lately, though, the "Remember me" box is unchecked every time I go back to these websites and I have to log in again. I made sure that Firefox was accepting third-party cookies, and I checked to make sure that it actually is storing cookies (which it is, from SeN I have PHPSESSID and life and it's still not logging me in automatically). What could the problem be?

Tools > Options > Privacy > Always clear private data on exit?
Or any similar security add-ons perhaps.

Always clear private data isn't on. As for add-ons, I have Stumble, AVG anti-virus, and a Microsoft .NET Framework thing. My computer was broken and it only started doing this after my dad fixed it. He didn't really describe what was wrong with it or how he fixed it, he just said that he switched the CPU out, messed with it, then switched the CPU back and it worked.