What does that mean? Basically, you operate on the mailbox that resides on the server as if it were local to your computer. Messages are not downloaded and deleted immediately, but reside on the server. There, they can be marked with several flags, for example "seen", "deleted", "answered", "flagged", or other, user-defined flags.
What else do you do with messages in your email client? You would file them in different folders, and you would search the folders for specific messages. Both can be done via IMAP on an IMAP server as well. You can set up mail folders and file messages there, and you can tell the server to search its repository and deliver the results to you.
IMAP also allows access to shared mailboxes. This is an easy means of sharing information, or to make sure an email inquiry (to a support mailbox, for example), is dealt with. All support staff can access the IMAP mailbox, and they will instantly see which messages have been answered and which are still pending.
IMAP was designed for online email access. You operate on the mailbox on the server as if it was on your computer. This allows for using multiple computers to access the same email account easily.
Example IMAP Usage
Imagine Jina, who loves working in the kitchen using her laptop and at the lake with an iPad but also has a computer at work.
When she had a look at her IMAP Inbox just before she left the office, there was an urgent message from John, her boyfriend. We don't know what he wanted to know, but it was important enough for Jina to flag the message as being urgent and waiting to be replied.
Coming home, Jina had already forgotten about John's message. But, thanks to routine, she dragged her portable computer to the kitchen table like she always did and checked her IMAP account's inbox. There was John's message again, demanding attention with its glowingly red flag. She replied immediately.
The message Jina sent back to John was automatically stored on the IMAP server in the "sent items" folder. On the next day at the beach, Jina's mailbox contained a message from John flagged as "answered" and a message from her to John in the "sent items" folder. She moved them both to the folder dedicated to John.
(Updated June 2011)