While the other answers are the "right" way to do it, they require some networking know how to get them working correctly.
If you want something that "just works" and you're willing to take a small performance hit on the new link, you can just buy a pair of powerline Internet modules.
Plug one into a wall power outlet near your router and run an Ethernet cable from it to one of the ports on your router. Plug the other one into an outlet in the "new" room near your other equipment.
If you just have one device in the new room, then you can run an Ethernet cable from the module directly into the Ethernet port on the device.
This will work without any configuration unless the two sections of the house are electrically isolated in some way - usually not the case. But don't plug either of the modules into a power strip, etc. which has an electrical noise filter in it. That could filter out the Internet signal.
If you have more than one device in the new room, then you will have to install a switch between the second module and the devices. That should be pretty simple to do, but I have never worked with switches, so I don't know how to configure them.
You might also be able to get additional powerline modules if you just need another port or two. I know this can be done, but I have never tried it. I have also seen at least one model where the second module had more than one Ethernet port on it.
I'm writing this with my notebook hooked up to a 500Mbs powerline Internet adapter because my internal WiFi card is dead. It works great, but I don't play any online games or do anything else which requires maximum bandwidth.