Using this site, the IP address for both my laptop and desktop comes up the same and does not change each time I check it. I am, by no means, educated in this area and am struggling to determine what this means (and/or what I am doing wrong).
The IP you see on our home page is your external IP. Your router assigns your internal IP's and routes the correct information to the correct computer.
To see your internal IP address's, go to a command prompt (Start > Run > type cmd and hit enter) In the black window type ipconfig and hit enter. your internal IP will be displayed in the list under your network adapter.
Most home routers are set, by default, to use a service called Network Address Translation (NAT) to provide Internet access. It takes you internal traffic, assigns the router's public IP as the source, and then tracks the return traffic to send the response to correct inside IP.
So, as Shnerdly mentioned, you have the IP that a site like us sees when you connect to us. That's the Internet IP of the router on your LAN. And then you have an internal IP (in a home network, usually assigned by the router).