A modem is always needed, because this translates the internet signal from your phone socket.
A router just splits this signal up for different computers to use. Wireless routers do the same as well as broadcasting the signal like a radio, for other computers to pick up.
So the line goes: phone line -- modem -- router -- computer
Many ISPs send out modems that connect via USB into your computer (I have no idea why). Most routers have ethernet inputs (ethernet is the standard network topology. ethernet cables look similar to phone line cables). so the problem here is that your modem is outputting the signal with USB, and the router is expecting it from an ethernet cable. the two aren't compatible.
solution: buy a modem+router in one. these are just one box, look just like a router, but have the modem built-in too, so you don't have the above problem.
there are two main types of broadband in the UK: ADSL broadband (down your regular phone line), and cable broadband (via a specialist fibre-optic line). Make sure you buy a router with the right modem in it.
Here's an example of an ADSL modem+router in one:
http://www.ebuyer.com/UK/product/97601/rb/2090 1461963
Finally, to receive a wireless signal (wireless internet on your laptop for example), you'll need a wireless receiver. Some laptops come with it built-in (many new ones now). If yours doesn't have one, you'll need to buy one. The above linked example has one packaged too.