I could be wrong, but it sounds like your wireless router is physically connected to your desktop, in which you're looking to replace with a laptop, is that correct? If so, then yes, it will absolutely work. Do you have any other wireless devices in your home--like an iPad, nook, kindle, smart phone, etc? You would connect your laptop via wireless the same way.
In a nutshell, here's how it works: "The Internet" comes into your home, and terminates at a modem. Modems usually only have 1 ethernet port, which is fine if you only want to have 1 device connected to the web--that means no wireless devices (ipad, etc), and only ONE desktop PC. In today's world, that's not really a viable solution. So, you get a device that accepts...