You can't check your IPs through any site on the Internet. Your router is using Network Address Translation, which allows all the computers on your network to share the single public IP assigned by your DSL provider. That means that all websites see only that one IP, no matter which of your computers is the one actually viewing the site, and the router handles the mapping and remembers which computer made the connection to that website.
The IPs that your computer have are usually handled by DHCP, dynamic IP assignment, by your router. More than likely that's how your network is set up, since it doesn't sound like you'd know enough to manually set up IPs. You can see what IP your computer is assigned by opening the properties for your LAN connection, and looking at the Support tab. It will be something like 192.168.0.100. Normally the router keeps track of which IP was assigned to which computer, so that it only is assigned to one machine at a time, however sometimes it can get confused.
If you restart the router, each of the computers should detect that their LAN connection is down, and when the router is back up they will retrieve a new IP address, and it should take care of any problems. It's possible you may need to reboot one or both of the computers.
