Yes you need a router. It needs to be a wireless router.
The signal that comes down your phone line goes into a modem. This can normally only serve one computer (via a cable).
If you want two or more computers to use the same broadband connection you need to buy a router. A router contains a modem so it replaces the existing modem.
Your internet supplier may supply you a modem for free, or at least sell you one that has a setup disk so it sets it up easily for you desktop and laptop.
The desktop PC can continue to use the modem/router via a cable, but if you buy a wireless router your laptop can connect via wi fi.
Any more questions just ask.