Yes all the laptop needs is a wireless adapter.
However, the router will usually need to be configured to allow access for the laptop, the newer routers have tight security. The laptop will connect at the speed the main pc is connected at, but will share the speed. So if you have a 1mb connection, and you download data from the internet at the same time, both will achieve a max of 64k. If one only is downloading then this will achieve 128k.
I hope this helps