In the musical world of MIDI, Velocity (sometimes called Key Velocity) is a value between one and one-hundred twenty-seven that describes how forcefully a key is pressed. What it means in practical terms is a bit complicated.
When a key is depressed a sensor detects its velocity. The velocity is then interpreted by a synthesizer so that a low value will usually produce a quieter, mellower sound and a high value will sound brash. While volume does play a part in this it's not really the point - the object is to create a more natural sound by mimicking the tonal qualities of an instrument rather than merely changing the decibel level (although it IS possible to make the changes solely one of volume).
Velocity also plays a part during MIDI playback; it's part of the note value that (along with pitch and duration) gets sent from a computer or sequencer to the synth, determining the eventual sonic characteristics.
Strictly speaking, velocity is nothing more than a variable that can be interpreted to mean whatever the hell you want it to - if you want the synhesizer to play a completely different sound for each of the 127 different velocity increments, you can do it. If you want it to sound like a piano when played softly and a fretless bass when thumped, you can do that to. If you really want a mindfuck, reverse the progression so that pounding the keys makes your speakers whisper while playing gently blows them out. It's an imperfect but an extremely flexible and valuable tool.