Two types of broadband:
-cable broadband, same as cable TV
-ADSL broadband, through a phone line
You probably want ADSL broadband (from BT, Tiscali, or most other providers), and so need a working phone line. You may need to phone BT and get it connected (paying some fee), and can get a type of line that only allows for very limited calls, but also broadband (last time I used this, about a year ago, it was about �12/quarter).
Once you have a phone line and phone number you can get broadband from BT or Tiscali or anyone you like. They often send you a modem, and some even send you a wireless router now too.
If you don't get a wireless router as part of the package then you'll need to buy a wireless router and modem package, like this one:
http://www.ebuyer.com/UK/product/118010
Then you'll also need a wireless laptop. If your laptop is quite new it's probably already wireless. If not, then get a 'wireless dongle'. The item I linked to above comes with one (only �2 more as it's on special offer so worth it).