11-04-2011, 12:50 PM
Apologies if any of this doesn't reflect your setup I'm filling in some gaps....
Your router is the device that, in most cases, creates and manages your network be it wired and/or wireless. In your case this is your Airport Extreme (guessing it's an extreme - white box - looks like your mac mini, rather than an Airport Express which is a quarter that size and plugs straight into a plug socket.
Either way, if you have cabled internet, your router will connect to a modem then to the cable box leaving your home.
Your router, presumably, already manages a wireless network that your laptop connects to OK.
You appear to have installed the dlink adapter onto your Mac Mini. Now all you should need to do is go to the wifi icon (top-right near the clock) and click it. After a few seconds the name of your wireless network (that your laptop connects to) should appear. Click on this and enter the password.
That should get you connected.
I've made many assumptions here. Let me know where it differs for you.