There's no such thing (specifically) as an internet connection to an email service. There's an internet connection from the exchange to your router and then one or more to your PC and/or other internet-enabled devices. If you're able to post here on AB using your home internet connection (rather than, say, through your mobile phone signal) there's absolutely nothing wrong with your router. The problem must lie with either the email service itself or with your web browser (if you're using webmail - or with your email app if you're not).
So what's actually wrong? Is it that you can't get any internet services at all through your router or is it only your email that's being troublesome? If it's the latter, what type of device are you using (PC/laptop? iPad? Mobile phone? etc), what method do you use to connect to your email (webmail or a dedicated email app?), which browser/app is it that you're using and which email service (Gmail, Yahoo, GMX, etc) do you use?