First check the plug where the network cable plugs into the desktop pc, when the cable is plugged in there should be 2 lights on. If the lights are on the NIC's plug shouldn't be an issue.
If lights are on open a cmd promt and type ipconfig, you should have an ipv4 address beginning with 192.168.x.x if you are on a regular home network. The default gateway should also match the first 3 numbers. Also try to ping the default gateway, the default gateway should be your router's ip address which is typically 192.168.1.1 or 192.168.0.1 but they do vary.