I'm probably a little older then most in here but here is what I remember:
Quad was short for Quadrophonic.
A lot of albums and tapes were cut in quadrophonic (4 track)
In car audio... Quad goes back to the first cars that had both front and rear speakers.
Then they created Quadrophonic stereos that had a control paddle (sort of like a joy stick) that did left / right balance and front / rear fade.
You could move the paddle around in circles making each speaker play independant from the others as fast as you could move the paddle around in a circle (sounded kind of cool).
In the center position, all four speakers played at the same time.
Kids would get high, drunk, trip... what ever and blast the music and then spin the music around the car from speaker to speaker (kind of trippy itself).
Quad: 4 speakers
Quadrophonic: Control of all 4 speakers independantly.
Here is a link that will explain it better:
http://www.rdrop.com/~dano/tunes/quad.html