Wi-Fi is just a technology that lets you send data via radio so no cables involved.
When you see the BTsomething that is probably your next door neighbours internet, you can't use that (unless it's unpassworded and you're a theif).
To make use of your current broadband wireless, you can do one of two things.
1. Set up internet connection sharing on a PC that is connected to the net and install a Wi-Fi card so that your laptop can connect to that and share the net. The host PC will need to be on all of the time though, well.. not all of the time, but will need to be on when you want to go online with the laptop.
2. Buy a wireless router. I'm prettery certain that you can get wireless routers that you just plug in your current net connection and then connect to that with your laptop. Any pc can be on and you will still be able to go online.
Just keep in mind that Wi-Fi's range isn't marvellous, you won't be able to sit with your laptop miles away and use your net connection. You can, however, sit in the garden and just about anywhere around your house. Oh, read about wi-fi security if you get it, too. VERY IMPORTANT!
Wi-Fi hotspots might interest you, too. They're in places like Starbucks where you can go in hook up to the net, browse a bit, check email, pay for how much you use and leave.
Hope this helps =)