I currently use a linksys router for my home network. I have one wired
ethernet cable running through my attic to a remote part of my home.
That cable is connected to a networked ReplayTV (DVR like Tivo) box.
I'd like to add another device to the network at the same location but
don't want to run another ethernet cable and I don't want to set up a
wireless bridge. Is there an easy way to share that one ethernet cable
so that both devices are connected to the network? If a hub or switch
is the answer, how is that configured, what type should I get?

You need a simple network switch, such as a Linksys EtherFast 5-Port
10/100 Workgroup Switch. No additional configuration will be
necessary. Simply plug the cable that is currently plugged into your
DVR into this new switch, and then you can have up to four outputs:
one for your DVR and three more for whatever else you would like.

You can probably find the hardware you need at a store such as BestBuy
or OfficeMax, Staples, etc.

that sounds perfect - thanks for the comment!

you will need a switch to connect in the attic. you will have to re
crimp the cable from one end to make a crossover cable to connect the
two hubs/switches. You will require one more cable to connect the
other planned device from the new switch. this would be a straight
cable

just find a switch that has an uplink port or does auto crossover
there is no configuring with switches just plug and go enjoy!