I guess Airplane mode should be OFF and WiFi should be ON for the hardware to be powered up and able to transmit a signal. And your router will usually be set to broadcast your SSID so your laptop can see it :emotion-1:.
Win 8.x should be the same/similar to Win7 as long as you are in the 'normal' PC mode and not in that idiotic single window tablet mode.
So from Win7, and within the wireless network icon on the notification area of the taskbar (normally on the bottom right) you shoud be able to left click and select 'connect'. All viewable SSID's should be displayed and you should be able to click on your network. The system will ask for the password associated with your wireless network (and you know you should have a password set don't you) and there should also be a 'remember' or 'always use this network' button which you can select.
Now if you don't broadcast your SSID, which seems to be the case, the procedure is the a little different but there should be a link you can left click to 'connect to a hidden network' where you can enter the SSID and password along with the 'remember'/always use' buttons. You should be able to work out the rest.
Don't discount that the wifi adapter may actually be faulty (DOA -dead on arrival) as it's new and we don't actually know it works....
Oh, and do check that the WiFi is actually switched on within your router as you do need to have a wireless network to actually connect to 🙂