When you retrieve your email messages from your internet service provider's mail server in your email program, the server and your program (the client) have traditionally used the Post Office Protocol (POP) to talk to each other.
Providing a way to get email messages to your computer is what IMAP and POP share. While POP was designed to do merely that, IMAP does provide more useful functionality, though.
POP, Offline, and the Problem
In a typical POP session, you will download all (new) messages and delete them from the server immediately. This procedure works perfectly fine as long as you access your email from only one computer (and only one email program).
As soon as you try to work on your email from more than one machine (a desktop at work and a laptop at home, for example), things get clumsy and complicated. Soon enough, you'll start forwarding and redirecting messages to yourself or set up two independent email accounts that forward to each other. Maybe you'd also try another esoteric, but not very elegant approach.
At the root of your problem lies POP's concept of offline email access. Email messages are delivered to the server. Your email program downloads them to your computer and deletes all messages from the server immediately. This means they are all local to your computer, where you operate on them.
Now how can IMAP help remedy this unfortunate, yet unfortunately not unlikely, situation?