When connecting computers to a router, can I use a common wire and then split it at each end to connect to two computer and into two ports on the router? Or does each computer need a dedicated wire?

No splitting. Use standard patch cables. One cable between router port and wkst nic.

Actually each connection needs a pair. It is possible to split the 4 pairs in a standard cable, but it depends on how far apart is the split. A couple of inches may be OK. There will be increased signal losses and noise pickup. I might try it a 10 mb connection but not 100 mb. Actually cable is not that expensive. The additional work in rejacketing the split ends would exceed the cost of another cable. 300 feet for less than $25. It would be easier to put a hub/switch at the computer end to split the signal.

I was just seeing if I could avoid the hassle of wiring along some of the same path twice. If it can't be done, then two wires it is. Thanks for the info.

Or as wizard-fred suggested at the end of his post add a hub/switch at the end of the wire. In other words: 1 long cable from the router to the switch. Then two cables from the switch to each computer. The switch is basically a hardware splitter. Michael J

Does a switch have any impact on the performance of the network?

No lost of performance. I use this method, a bridge out to upto 3 switchs, thay also give you a repeater effect if on a long run.