The certificate is not trusted because no issuer chain was provided.
(Error code: sec_error_unknown_issuer)
So I am fairly certain that fixing certificate chain will make it work even on older browsers, since the root certificate is long time since valid (and would be found in almost every browsers CA store) as opposed to “Rapid SSL CA” intermediate certificate, which is relatively recent (Valid from Sat Feb 20 00:45:05 EET 2010) and might not yet be redistributed to browsers (which would be unnecessary to begin with if only everybody configured their servers properly )
I get an error on the home page with both IE and Chrome.
Seems that you have resources in this page that are not secured. Make sure all resources are embedded using SSL (HTTPS) and not plain HTTP. Otherwise, browsers will show you an error.
I dont get an error on the logo. It could be that your browser is not updated or that you are using a weak cipher. As Aleksejs suggested, you should look at the specific error generated. In IE9 you should see and error on screen, in Chrome, click on the red icon next to the URL.