The dialog shows up for HTTP "Basic" or "Digest" authentication. This should not be used for anything serious. The "Basic" authentication sends passwords in the clear to the server. "Digest" is somewhat better, but there is no way for a user to detect whether the password he's supplying will be used for Basic or Digest (it might not even be clear to the user whether he's authenticating with an HTTP or HTTPS server).
Finally, most browsers offer only very obscure ways to clear a password for these authentication methods once it has been typed in. There is no way for the website itself to force a "log out" purge. So users who need to use shared/public computers will not be able to log out afterwards.