You are not going to see the , lets call it the WE, because you don't have a wireless card in your pc. Your router is not set to bridge mode and you dont want it to be from what I gather. I'm assuming from what you say that your pc is connected by a network cable to a wireless router which is connected to the internet.
If that is correct then to install the WE set it near your router, give it power, and turn it on. Let it boot up (usually takes a few minutes)
With your computer off unplug your computer network cable from the back of the router and plug it into one of the ones on the WE.
Boot up your computer and it should find a new network. let it run thru its configuration and then open a browser and go to WWW.MYWIFIEXT.NET
Log in with admin using netgear as the password.
Using the menu on the left, select SETUP WIZARD and then find and connect to your routers wireless network. You should already know your wireless routers connection info (ssid & password) and once you connect you will get a 'Congratulations' screen.
At this point you can click finish and be done but I strongly recommend you update the devices firmware (the link should now be available on the left side now that you've connected to the router) . Once thats done you can power down the WE, reconnect your computers network cable back to the router, and then relocate the WE to someplace in between your wireless router and the area you want people to connect in. Power the router on amd once its booted up fully check the connection light, ideally you want it to be green because the stronger the signal the faster the link will be (ie: green = more speed) This link light is showing you the connection between the WE and the router only, not the pc's connecting to it.