To make use of your home mac mini server, you're going to need a couple of things...
The first is probably going to be a dynamic dns updater (since your cable modem IP address will likely change). Here is one: http://www.dyndns.com/index1.html
Either that, or call your ISP and ask them for a static IP address. They might charge you a small fee for this (ask to talk to tech support for this).
After you have that, you'll need to log into your router and set up something called port forwarding (I apologize if this sounds obvious. I promise I'm not trying to talk down or anything like that, just trying to cover all of the steps).
Forward port 80 (it might just be called "WWW" or "HTTP" in the router settings. This is how it is on one of the sonicwall appliances we use at work) to the internal IP address of your mac mini server.
Make sure that you set a static IP address for the mac mini. You can do this by going to system prefernces -> Wireless and Networking -> Network -> Advanced -> TCP/IP -> select "manually" from the dropdown menu and fill in an IP address, subnet mask (probably 255.255.255.0) and a gateway.
Once you have port 80 forwarded to your mac mini, and a DNS record set up, you should be able to access your mac mini's web server from the rest of the internet.