Hi,
I have a desktop computer at home connected up to Telewest Broadband Cable Internet. I am wanting to make my laptop connect wirelessly to the internet. If I buy a router to plug into the cable modem, does that mean my laptop will connect wirelessly WITHOUT having to turn the desktop computer on?

thanks,

Craig.

Yes
All you need is a wireless router and a wireles card for your laptop if it does not have built-in wireless capability.
you can plug your modem into wan port and your PC to lan port of the new wireless router,
you can allways have wireless connection to your laptop without the main PC switched on.