What’s the difference between Live mode and Quick mode autofocus in Live View on a Canon 550D?

There are two different types of Autofocus in most DSLR's. There is PDAF (or phase detect autofocus) and CDAF (contrast detect autofocus). PDAF uses a special focus sensor to measure the phase of the light coming from different parts of the lens (which is much faster) while CDAF uses the actual image sensor to look for the highest contrast it can get (which requires moving past the point of ideal focus and then coming back to it, which makes it slower.)

While some recent DSLRs have come up with ways to include PDAF sensors on the main imaging sensor, many models still use a completely separate sensor (either for cost or performance reasons). When you are using Live View, the mirror is flipped up and thus no light is going to the PDAF sensor or the view finder.

Normal live view focusing will use CDAF and hunt for the proper focus, which can have an advantage in certain low light situations and be more accurate in some cases, but it is also much slower. (Things like face detect are also possible depending on your camera model.)

Quick focus on the other hand will flip down the mirror so that the PDAF sensor can be used, make the necessary focus adjustments and then snap the mirror back up to take the photo. It is important to note that the image doesn't actually capture when you hear the first mirror movement, but rather after the second mirror movement.