I just tried to log in using IE8 and was able to register without a problem. The pop-up came and went and I was redirected to my account in JomSocial when done.
If that doesn't work, I'd recommend clearing your browsers cache and trying again.
Finally, can you try with a different Facebook account, or on you Facebook page, go to Privacy Settings and in the left click on your applications. On the new page actually delete the application for your page. This will remove the permissions you've given for that app to read your information. Then, try and log in again (you'll get a new permissions pop-up). Hopefully that will help.
Sometimes, the cookie that Facebook sets can get wonky and cause issues like your seeing, and there was a Facebook Connect issue in general late last week that could have caused this. Hopefully one of the above will solve your problem, but I think it's something localized to your Facebook account on the site. Can you try logging into this site, or the
Joomla Facebook Demo
site and see if you get the same issue there.
All of this will help us narrow down the problem.
Thanks!