The Annoyance:
We have two computers in our family room, connected to a router that sits between the two computers. My wife wants to add another computer to the network and put it in the den, which is next to the family room. What's the easiest way to do this?
The Fix:
I'll tell you the easiest way, and then I'll tell you the best way. The easiest way is to string the cable along the baseboard, or below the quarter round, and take it through the opening (under the door if there's a door) and into the next room. This is the easy way because it requires no tools.
The best way is to drill holes and string cable through the holes. This method uses less cable, and in the end, the holes are less conspicuous (and less ugly) than the cable. Drill a hole in an inconspicuous place. I usually pick a corner of the room and drill the hole right above the baseboard. The hole only needs to be slightly larger than the connector at the end of the Ethernet cable, which is about half an inch. Do the same thing in both rooms (measure from a corner or the doorway, so the holes are directly opposite each other).
Cable is floppy, so when I'm cabling adjacent rooms I tape the cable to a thin dowel (I keep the chopsticks I get from the local Chinese restaurant takeout counter for this purpose). Duct tape is great for this type of work because it doesn't come loose if you scrape against the wall. Electrical tape works almost as well. Shine a flashlight through the hole from one room so that you can see the opening clearly, and push the cable through from the other side.
Don't put tape on the Ethernet connector. You don't want to get sticky stuff on the business part of your cable.