No internet on one cpu but not the other

check that you have dhcp enabled on the router and have not set a manual ip on the desktops connection.

im guessing the router will have dhcp enabled already, so just check the desktops connection settings by right clicking the device in network connections (via control panel) and selecting properties, then properties again, select the option in the window for tcpip and select properties, on the window that comes up the tick box for automatically find my ip should be ticked, if not, first write down the settings currently entered on a bit of paper and then tick the box.

select ok a few times and the connection will then look for a new ip from the router and hopefully get online.

Try manually assigning the PC the IP addresses.
Just don't assign the same IP as the laptop is using.
You need :

IP address
Gateway IP address
DNS IP addresses

Run an "ipconfig /all" command in a CMD window on the laptop to determine what addresses to use.