Pentium100's answer is quite complete, but I'd like to give a much simpler (though less accurate) explanation.
The reason because sinewaves have (ideally) only one harmonic is because the sine is the "smoothest" periodic signal that you can have, and it's therefore the "best" in term of continuity, derivability and so. For this reason is convenient to express waveforms in terms of sinewaves.
These images explain better the square wave decomposition in harmonics:
AM modulation is analog stuff, not "on" and "off", and basically means multiplying a signal for another one, to raise its frequency. So you (ideally) don't create harmonics, just shift the existing ones to higher frequency.