A few pieces of information from you might be useful here:
Firstly, how is it apparent that the laptop isn't connecting to the internet? If it's simply that web pages won't load in a browser, it could be a problem with that browser, rather than with the laptop or router. However if 'no internet access' is shown when you look at the available connections (via the icon at the bottom right of the screen) then we can rule out the browser.
Next, have you tried connecting the laptop to the router with an ethernet cable? (The router should have been supplied with one, or you can hand over a quid in Poundland for one). If the laptop works with an ethernet connection it's clearly a wi-fi problem; if not, we need to consider something else.
Have you tried taking the laptop into your nearest McDonalds, Wetherspoons, Tesco Extra or Asda to see if it will work with their free wi-fi connections? If it does, the problem would seem to be with your router, rather than the laptop.
Lastly, have you got any other devices that take a signal from the router (such as another laptop, router or internet radio)? Are they working normally?