Solution:
Restart IIS. In most cases this is all that's needed.
If that doesn't fix it, there is probably a permissions problem that you will need to resolve. FogBugz is trying to write to the registry on your server, and it does so via the user account FogBugz is assigned within IIS. Try to log on to the web server as that user. If you can't, that account is not a valid account. Once you have an account with which you can log on to the server, tell IIS to use that account for FogBugz, and that should fix it.
Normally this FogBugz user account is the account used for anonymous access in IIS, but if you have explicitly set IIS to instead use Windows authentication for FogBugz, then here's what you need to do: all your users should be members of a Windows user group, and that group needs to be given full permissions on the registry key in question: HKEY_LOCAL_MACHINE\SOFTWARE\Fog Creek Software\FogBUGZ\
If further troubleshooting is needed, you could use a program called Regmon to see what's going on at a low level, this will show you any permission problems.
Update:
If it looks like the permissions are right and you checked the identity in the site and it looks right, IIS6 has the identity set in both the website and the application pool. Make sure both are set correctly!
