I can’t put it any better than @aprilsimnel did. It’s the address, plain and simple.
You don’t drop a blank envelope in the mailbox and expect it to go to your grandmother. You write her address on it.
Every packet of data you send has the recipient’s IP address, and every packet of data you receive has your IP address on it.
Without the IP address on it, it wouldn’t get to its destination.
They’re completely independent from email addresses, but they are both addresses after all, so they work similarly. Just, differently.
edit: A packet is any piece of data to be sent over the internet, whether it’s an IM, an mp3, a myspace update, character movements in computer games, html files, etc. etc.