What all do I need to install a second desk top in another room in my house?

If you have a router connected to your modem and the router has more than one port for the internal connection, simply run the ethernet cable from the new machine to the router.

If you do not have a router,. then purchase one. Just about anything will work, but make certain it offers at least 100mbit internal connections if not 1gbps. You will need to run cable from the router to both machines. I recommend purchasing CAT6 cable, but do it from the internet, like Amazon or such and you will be able to get it for a great deal less than with a local purchase. Online I have purchased 25 feet for under$10 while the local store wanted $40 or more for that same 25 ft.

If you have a router which has only one internal plug, you can either replace that router with another (most home routers have 4 internal ports) or you can purchase a switch. Interestingly, switches are usually more expensive than the home routers even though the routers do so much more. Basically, a router is a switch with a bunch of server functions built in to allow for traffic leaving the network versus the traffic on a switch being just for the local connections.

an easy to install router or ethernet splitter

A router.