An IP address is like a telephone number - its an individual number that identifies a piece of hardware connected to a network, whether that be a private network in an office or your computer connected to the Internet
IP addresses allow Internet connected computers to talk to each other, something that we depend on when visiting web sites and checking our email.
We all know when we want to read the news, we point our browser at, for example, the BBC or CNN webpages. Behind the scenes, computers, routers and web servers are communicating using IP addresses - domain names are the human way of making the computers chat together nicely.
What would you rather try to remember, http://212.58.224.138 or http://bbc.co.uk. As humans we are more tuned to words and not numbers!
Your IP address can identify you wherever you go on the Internet, visiting a website will mean your IP address and your browser identifying string (the 'user agent') are stored in the log files of that website, when you send an email your IP address will be in the headers (behind the scenes) meaning anyone who reads those logs or looks at the headers of the email address can see your IP address.
If you are participating in illegal activities on the Internet, from threatening someone in an email, through defacing websites to downloading illegal material, law enforcement agencies can get access to your IP address and then approach your ISP to see who was using that particular IP address on a particular day. If you are in too deep, with the information gained from just your IP address could mean an early morning call from the police knocking on your door!