I assume it's not this side (forums.cnet.com) but some other site that you call "this site".
So it doesn't work in Firefox on your PC. Alas, the only one that can answer your question "Is it the site or my browser that has the problem?" is you. It could even be your PC.
To answer your question:
1. Try it on your PC in another browser: IE, Chrome. Works? Then it was the browser.
2. Try it on another PC (friend, family, work, school, etc). Works? Then it was your PC.
3. Works nowhere? Then it's the site.
Simple recipe to get the answer to your question.
I'm a member of this site who's recently not been able to log in even though my username and password were correct. Site says to clear the browser cache and cookies (Firefox), which I did repeatedly, but nothing happened (can't even log in the forums to ask for help). Tried requesting a password change but still nothing works. What's going on? Is it the site or my browser that has the problem?