Your LAN card aka Network Interface Card either isn't setup or you don't have it set up properly for TCP/IP protocol. You using XP? Win98? Other?
If you added a NIC, then load the driver files for it. If the NIC is built into the motherboard, then load the correct motherboard drive files.
It's basically the same for all. You need to go to the networking section such as in Control Panel at the networking icon and make the correct settings. Open it and make sure the LAN card is showing as an adapter, that a TCP/IP protocol is loaded pointing to that LAN card, and there's a client such as Microsoft Networking. Those three. If not, take corrective actions.
If you see those then open properties on the LAN adapter shown there and make sure it's bound to the TCP/IP.
Open properties on the TCP/IP protocol and bind to the clients, obtain IP automatically, use DHCP, don't use WINS, disable DNS, leave gateway blank, no NetBIOS needed. In the network icon on another tab make sure you have a computer name and workgroup name. You will want all computers to use this same workgroup name if you later want them to see each other for Sharing.
Do that on the desktop and then check to make sure it's also same on the laptop, since you have two problems here. One problems is connecting your desktop with a LAN to the router, the other is connecting wirelessly with the laptop.