Indications are that there is a problem with the wireless connection. But not sure whether it is at laptop or router end. It is odd not to lose email if the internet is not working. How long has this problem been happening? It may be worthwhile doing a system restore to just before the problem.
In your Firewall check that you have given Firefox permission to connect to the internet. It sometimes happens after programs (firefox in your case) have checked for updates if you haven't got them listed as ok in your Firewall list.
Yes, I'm confused that email works but browser keeps losing connection.
It's my wife's laptop, which I've only recently started using. Because of this, it has had very little installed in the way of software. It has Norton Internet Security, as does my laptop, and only being used for a bit of browsing & internet banking - plus playing a few games, there has been no need to keep backing it up.
She always complained she couldn't access her bank account, and my answer was always disconnect & connect again, which would get it working long enough for her to do what she wanted. Now I'm using it, the problem is rather annoying, and I can't really go back to a point to where it was working properly