If you have open WiFi, or it's only secured by WEP, a neighbour can very easily illegally download and shared copyrighted files over your broadband connection. A copyright holder can detect this (usually by connecting to BitTorrent trackers to find out who is sharing the files), and get the IP address in question.
They then find out the account holder's name and address from the ISP, and today they will send them a letter, usually saying pay us something like £500 or we will sue you. Note that this goes to the broadband account holder, not the neighbour.
The new part under the proposed Bill is that the copyright holder can get you disconnected from broadband if they send you a number of warning letters and the filesharing does not stop. There are various steps they have to go through - see
http://www.dontdisconnect.us/the-proposals/ - and if you can stop the neighbour doing this, ideally by securing your WiFi, you should not get disconnected.
If you keep getting the warning letters (perhaps because of guests, children or you don't know about securing your WiFi), and don't win an appeal, then you will eventually be disconnected from broadband - or at least that's one of the "technical measures" that is being considered in "secondary legislation". One of the problems with the Bill is that once it is law, there is a lot more flexibility for the government to implement measures that it wants, with less scrutiny by Parliament than a normal Bill. So it's hard to say exactly how long a disconnection would last, or whether other measures would be taken first (e.g. traffic shaping to a very low rate).