I was wondering!! I like configuring my router so that I can access it over the Internet even when am not home. What are the most critical risks if a hacker got my router password?
There are a few main things. The first thing would probably be to take down your firewall and expose your internal network to the outside world so that they could access your internal traffic.
The next would likely be to change the DNS information so that it provides a misbehaving DNS server that could direct you to fairly convincing phishing sites. While SSL locks wouldn't appear, when you went to www.mybank.com, they could send you to a site that looks just like mybank.com but doesn't have any SSL and try to steal your login.
What else they could do depends largely on the capabilities of your router. If VPN is supported, they could actually connect directly in to your local network and monitor traffic between your systems, potentially gleaning significant information. Any systems with vulnerabilities due to lack of patching would be accessible to them.
All in all, it isn't like it automatically assures they will get access to anything, but it does mean that any amount of defense your router normally gives is gone and gives them control over how your network behaves.
(S)he can own your network. Your router is a networking device with an embedded operating system, usually some flavor of linux. At a minimum if you're popped by a script kiddie the attacker could change any settings you see on the web interface. If it's popped by someone who knows what they're doing they will essentially own a machine that is sitting at the point in your network where all data passes, sniffing your internal and external traffic, scanning your network, forwarding ports (s)he can have direct access to compromise any vulnerable machines on your internal network, setup vpn tunnels, change firewall rules etc...
I would strongly suggest not opening up that functionality to the internet. If you REALLY want to manage your router from over the internet I would suggest to setup a VPN so you can securely tunnel into your home network then access it from there.