MIME is an abbreviation for Multipurpose Internet Mail Extensions. Behind this rather cryptic expression lies an internet standard describing how to send messages containing data other than plain text across internet. For links to the standards themselves, read this topic.
Back in time, it was only possible to send plain text. MIME was invented to allow messages to contain enriched content, such as graphics, HTML, sound, video etc.
A MIME message, as opposed to a plain text message, contain one or more parts. Each part has a content type identifier, saying something about what the part contains. You have probably stumbled across content types such as text/plain, text/html, image/gif, image/jpeg or audio/mpeg. A simple e-mail reader can for example only read plain text messages, and will automatically choose the part with content type text/plain, while a more advanced e-mail reader would choose text/html in order to see the mail with more rich experience. One of the cool things with MIME is therefore the ability to incorporate different content for different readers. One thing to remember, is that imbedded graphics is always stored as separate parts, with content-types such as image/gif or image/jpeg. The MIME standard define the order of- and how each part should be linked together.
All modern mail clients can read and write (or de- and encode ...) MIME messages nowadays, including Lotus Notes.