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MIDI stands for Musical Instrument Digital Interface.
Musical Instrument Digital Interface is a now industry-standard protocol that allows all types of electronic musical instruments (keyboards especially) to communicate and synchronize with computers or other pieces of equipment.
Using MIDI, sound cards, computers, drum machines, synthesizers and MIDI controllers can each sync with each other while exchanging system data.
Instead of transmitting audio signals however, it instead transmits a digital stream of "event messages" that can control signals for volume, pitch, etc.
The original MIDI protocol was published in 1983 and the standard has gained widespread industry support since then and continues to be the height of unification for an industry.
The MIDI interface uses a connector that looks exactly like the one pictured at right. Each device needs a MIDI In or MIDI out port.
MIDI messages, along with any relevant timing info, is stored as a computer file, normally known as a MIDI file. The full term however, is Standard MIDI File (SMF).
An excellent place to find MIDI files for yourself that work on mostly all keyboards or equipment is the original MIDI site, available here: MIDI.com.