Your incoming internet connection - cable/dsl - connects to the WAN port on the router (which is a powered device), and the individual PCs connect to either the wired LAN ports (typically four) or wirelessly. So as long as your router has power you can connect. The general idea is to have the desktop connect using wire and the laptops wirelessly. Why? Convenience and the fact that the printer and other peripherals are already connected to the desktop, that can then be shared by laptop. Of course you cant print unless the desktop is on. Large data transfers over wireless are slow - so when you need to do that, you would connect the laptop to one of the free wired ports on the router.
When you buy that laptop make sure it has a wireless adapter (802g), and a network connection (most typically do) and buy a wireless router with it and two network cables with it. Setting all this up is quite simple.There are plenty of threads about recommended routers. I have been using a Motorola WR850g for two years- and in a two storey house only the patio has poor connectivity.