IP addresses is short for for "Internet Protocol Address". These are numbers assigned to points, like routers, or computers on the Internet. They are designed to let computers find each other across networks. Domain names are mapped to IP addresses to make them easier for humans to remember.

IP addresses can be fixed, or can be assigned when you are added to the network. Thus your computer may not always use the same IP Address.

Rights issues

IP addresses can be used by law enforcement
IP Addresses can be used by private individuals as a way to request details of an ISP's customers, if a judge can be convinced that criminal or civil wrongs may have taken place. These orders are called Norwich Pharmacal Orders
Under Data Retention, IP Addresses of users are stored for potential investigations
Nevertheless, IP Addresses cannot identify a signle person, only a machine. This makes them a coarse tool for identifying criminal or civil wrongs.

Wikipedia IP Address page