What do all of those specifications on audio amplifiers mean?

The term "Music Power" has been used in relation to both amplifiers and loudspeakers with some validity. When live music is recorded without amplitude compression or limiting, the resulting signal contains brief peaks of very much higher amplitude (20 dB or more) than the mean, and since power is proportional to the square of signal voltage their reproduction would require an amplifier capable of providing brief peaks of power around a hundred times greater than the average level. Thus the ideal 100-watt audio system would be capable of handling brief peaks of 10,000 watts in order to avoid clipping (see Programme levels). Most loudspeakers are in fact capable of handling peaks of several times their continuous rating (though not a hundred times!), since thermal inertia prevents the voice coils from burning out on short bursts. It is therefore acceptable, and desirable, to drive a loudspeaker from a power amplifier with a higher continuous rating several times that of the speaker, but only if care is taken not to overheat it, which is difficult, especially on modern recordings which tend to be heavily compressed and so can be played at high levels without the obvious distortion that would result from a 'real' recording when the amplifier started clipping.