There are two ways to do this:
1. Internet Connection Sharing (ICS).
2. A cable/dsl router (preferred, trust me).
A splitter will not solve your problem. Somebody either doesn't know the difference between a router and a splitter or is blowing smoke.
This is how you physically hook up a router to a cable modem:
1. Plug the cable from the cable modem into the WAN port of the router.
2. Then plug your computers into the LAN ports of the router.
If your router has only one LAN connecton, you will need a network switch to enable more than one computer to connect to the router. So I would look for a cable router with a built in network switch.
If you want to go wireless, then you need a wireless cable/dsl router. The hookup is basically the same to the cable modem. But you don't need to run wires to the computers from the router.
You will need Ethernet cards in both computers. If wired, then the one with the cable now will work as is. You will need one more card. If you go wireless, you will need between one and two wireless Ethernet cards. At least one for the distant computer. Possibly one for the computer with the current cable connection, depending on the configuration of the router.
We have one computer (downstairs) hooked up to a cable modem through charter communications and we just got another computer (upstairs) that is currently hooked to aol dial up. We would like to have the second computer on cable broadband also so I just called the cable company and the tech said oh we don't deal with that what you need to do is get a router or a splitter hold on while I check he comes back and says yes it's a splitter you need. I explained that I don't want wires throughout the house but this is what he said I need to do. This does not sound right to me I don't know if it is called networking or not but what I thought they would do is give us a wireless box for the upstairs. I don't care about sharing info with the downstairs computer I strictly want the broadband access on the second computer. I don't want to have to pay another $45 a month for them to hook a cable modem up here. Can someone please tell me what to do?