IMAP: acronymic for Internet Message Access Protocol. Unlike POP3, IMAP is designed for flexibility and for the organization and maintenance of mail at a central server - rather than "pull" email from that server and delete it, IMAP retrieves data as requested and allows users to create folders and move their mail around in an organized fashion from folder to folder on the parent server.
One of the major benefits of this model of mail access is that you can access old as well as new emails from anywhere you happen to be, and organized in precisely the same fashion. For example, if you have webmail installed, you will be able to see emails you have already read with a mail client as well as new emails which have just arrived with your webmail interface. By contrast, a POP3 user would probably have removed all already-read email from the mail server when they viewed it - meaning that only new emails would show up in their webmail interface - and even if they had configured their client not to remove email from the server, any organization of their email into folders would only have been at the client level, and so at the webmail interface the mail would just be one big jumble rather than being organized in the way the user had become accustomed to in his own client.
Another advantage is that backup can be centralized at the server level rather than needing to take place at the client level. And finally, it's frequently invaluable for a user to be able to "get his mail back" on a new workstation simply by setting up the account instead of needing to first remember to back up and then somehow migrate his mail from old workstation to new.
One disadvantage is that IMAP is still a fairly new protocol, and trades some ruggedness under adverse network conditions for its flexibility - you should expect to have to stop and restart IMAP mail clients fairly frequently, as most of them do not always recognize when an open network thread has been broken due to some network problem (or impatient router) in between the user and their server, and will endlessly try to re-use the broken thread rather than intelligently close it and open a new one. Of course, how often this is a problem depends in large part on how reliable your network connection is; and clients are getting better about recognizing broken pipes and reacting intelligently.
If you are willing to mollycoddle your mail client a bit, and have the server space to spare to keep your email stored there instead of at individual workstations, IMAP is highly recommended.