From MozillaZine Knowledge Base
IMAP (Internet Message Access Protocol) is a network protocol used to access e-mail messages while they are still stored on the server. It supports a client server view of the world. This means that IMAP lets you access messages stored in a remote folder as if it were a local folder. This includes copying, moving, reading, and deleting messages. Thunderbird by default also keeps copies of all of the messages in your remote folders in "offline folders", stored on your hard disk. When you are working online you see the contents of the remote folders. When you are working offline (File -> Offline -> Work Offline) you see the contents of the corresponding offline folder instead. You can disable this feature in Tools -> Account Settings -> Synchronization & Storage.
Most users are more familiar with POP (Post Office Protocol), which supports a download-centric view of the world, i.e. you read a message by downloading it from the server's inbox folder rather than remotely viewing it wherever it's stored. IMAP is a newer and more powerful protocol. It lets you use all of the folders (just like webmail) while POP accounts are limited to just the inbox folder. However, many ISPs prefer to support POP instead because most users are used to it and they can provide a smaller mailbox (since users will be downloading messages and keeping none or only some stored on the server).
If you have only used POP accounts you may be used to deleting a local copy of a message, knowing that the original is still stored on the mail server (and accessible via webmail). That isn't true with IMAP. That is why the folder in the IMAP account is called a remote folder. Whatever you do to the remote folder affects the corresponding webmail folder. It also affects the corresponding offline folder, if it exists.
Thunderbird, the Mozilla Suite, and SeaMonkey support IMAP accounts. Some of the main features are:
Many IMAP email providers provide additional functionality such as server side filtering, the ability to automatically fetch and merge email from several external POP accounts (or hotmail) into your IMAP mailbox, aliases, plus or subdomain addressing (which are usefull for creating email addresses on the fly to help deal with potential spam), and run server based tools such as SpamAssassin (which adds headers useful for spam filtering).
AIM (AOL) provides a 2GB mailbox, though its features are non-standard. Gmail has a 15GB mailbox, with some quirks such as a optional All Mail folder. GMX has a 5GB mailbox, with standard features. Yahoo and Outlook.com have standard features and supposedly provide a unlimited size mailbox. Outlook/Outlook Express, Thunderbird/Sea Monkey and Mulberry are considered the email clients with the best IMAP support.
IMAP is mainly used for mail messages but it's possible to use it for other types of storage. For example, one email provider also lets you store files in remote folders and configure web access to them. It essentially adds a WebDav server that accesses the files in the remote folders. You could use this to offer a home page for example.
Thunderbird specific features
"Push e-mail" means that when a message is delivered to the mailbox you want to get immediately notified of it. Thunderbird provides that if your IMAP server supports the optional IDLE command. The IDLE command eliminates the need for a e-mail client to poll for new mail - the mail server automatically notifies the e-mail client whenever there is new mail. "Push support" is sometimes mis-interpreted as IDLE support but it is actually proprietary Push-IMAP enhancements for mobile devices such as cell phones developed by Oracle. It can use IDLE, SMS or WAP Push to provide the notification.
The IETF has defined a standard called the Lemonade Profile that is meant to replace Push-IMAP. It includes other features such as the ability to forward a message without downloading it and the ability to quickly re-sync a connection. The Lemonade for Mobiles web site has more information on what e-mail clients and servers support Push e-mail and/or Lemonade. While Lemonade is mainly thought of for cell-phones it is also useful for laptops on trains and planes, and PC's using satellite links. Unfortunately it wasn't even considered for Thunderbird 3.
New functionality being developed
Discussions about a possible replacement for IMAP. Having the protocol also support address books, calendars and submission is being considered. Several people active in the tb-planning mailing list are involved, but no Mozilla employees.