MIDI stands for "Musical Instrument Digital Interface ". It is a set of hardware (IN, OUT and THRU ports found on electronic keyboard instruments) and software specifications for allowing electronic musical instruments (and also computers) to communicate with each other.
MIDI music is created using synthesizers and/or samplers to provide the sounds and generally, a keyboard or other MIDI capable device inputs the musical data to a sequencing program. The program records the information as text data, and the magical part happens when you press playback and the synthesizers duplicate your performance!
Fortunately most computers now support some way of playing MIDI files internally. QuickTime software from Apple, GS Wavetable for Windows and various plugins for Netscape and Explorer provide the basic "virtual sounds" your computer needs to hear the files. There are all kinds of MIDI players for free on the net.