A precision aerial maneuver; essentially, a maximum-performance climbing turn. The chandelle is one of the required maneuvers on the FAA practical test for the Commercial Pilot Certificate.
The chandelle starts with the aircraft in a straight and level attitude, at cruising speed and in a "clean" configuration (wing flaps and landing gear retracted, if applicable). From there, the pilot applies full power and rolls into a constant angle of bank (typically 30 degrees), and begins gradually increasing the pitch attitude until 90 degrees into the turn. After the 90-degree point, the pitch remains constant and the bank angle gradually decreases until the turn reaches 180 degrees, when the maneuver is complete and the pitch attitude is returned to level. At the end of the chandelle, the airspeed should have dropped to within five knots of stall speed (Vso). In short, the first half of the turn is performed at a constant bank angle with changing pitch, and the second half is the reverse: constant pitch, changing bank.
The point of the chandelle as a training maneuver is to demonstrate the pilot's understanding of aircraft performance and ability to control it with precision. Specifically, it places the pilot in a situation where several flight parameters are changing at once, all of which must be compensated for in different ways: rudder to counteract torque and aileron drag, decreased elevator pressure as the bank angle decreases, and so on. Doing all of this simultaneously requires both practice and theoretical knowledge, hence the chandelle's appearance on the Commercial flight test.
My main source for this is an old photocopy given to me by my flight instructor, from a now-unknown training manual. That and my own memory, which also tells me that when I first started doing "chandelles," I thought they sounded like a good name for a Motown girl group.