An ip address is TCP/IP's answer to a phone number split into four intergers. These are registered by your isp and administered by various non profit organizationd depending on where you live.
The first set of numbers is unique to your area (as stated by Thunder) and it's size determines the size of the network you're on (The smaller the first number, the smaller the number of nodes on your network). The larger your network is, the greater proportion of the ip address will be dedicated to your host id, which identifies you on that particular network. nIt's all pretty cleaver really

There are various ip analyzers out there that tell you details about a given ip address (
www.nic.com has one). The info you can get on someone once you have their ip address is impressive.
Most cheap isps give you a dynamic ip address (one that changes each time you connect). This can be a pain if you want to run any servers (in fact it makes it pointless: Think about what it would be like if your phone number cahnged all the time) so you can get a better, more expensive isp to give you a static ip address (one that stays the same all the time).