Ah, wireless networks, the bane of techies everywhere!
You haven't mentioned OS, so we'll assume you're on Windows XP.
Start
Run
Type in 'cmd' and enter. You'll be presented with a little black Window.
In it, type 'ipconfig' and enter. You'll see one, possibly more sets of numbers separated by dots. One of the sets will say something like wireless nework adapter (or it may call itself ethernet adapter 0, or another number).
One of the number sets will be called IP address. On most home networks it will start with 192.168, followed by two more numbers separated by dots. If yours is 169.254 then you have not successfully connected to the wireless network (or the router is not giving out addresses).
If this is the case you will need to manually assign an IP address to the laptop. Look up instructions on this elsewhere and make sure you give it the same first three numbers as the working computer (for example, if the working PC is 192.168.0.100, make the laptop 192.168.0.110)
However, I doubt this is your problem. The laptop is most likely trying to get online the same way it does when it's at home. Open up Internet Explorer (ignore the no page error) and open Tools - Internet Options.
Go to Connections
Select Never dial a connection
Click on LAN settings and ensure all boxes are unticked.
OK
Restart Internet Explorer.
Remember to go back into connections and select 'Always dial my default connection' when the laptop returns to its natural habitat!