Turn off the desktop, and connect the laptop to the wireless signal your router is putting out. Why even bother getting rid of the desktop, you can use both computers online unless you want to sell the thing.
I'm thinking there are some unasked questions you may have though. Read the router manual, will tell you how to hook up computers to it.
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 multiple inputs (usually about 4 physical ports and wireless functionality), like a wireless router. The router will manage all of your wireless devices in your home, then send all the traffic out through 1 ethernet cable to the modem, which is your "gateway" to to outside world.