Audio power
Audio power is the electrical power transferred from an audio amplifier to a loudspeaker, measured in watts. The electrical power delivered to the loudspeaker, together with its sensitivity, determines the sound power level generated (with the rest being converted to heat).
Amplifiers are limited in the electrical energy they can amplify, while loudspeakers are limited in the electrical energy they can convert to sound energy without distorting the audio signal or being damaged. These power ratings are important to consumers finding compatible products and comparing competitors.
Power handling
In audio electronics, there are several methods of measuring power output (for such things as amplifiers) and power handling capacity (for such things as loudspeakers). The question has engineering, regulatory (consumer protection and advertising), and psychoacoustical aspects and is, in a serious sense, much more complex than may be imagined.
Amplifiers are valued in part by their power output capacity. And in the interest of being able to advertise a higher power output number, manufacturers in the US (and elsewhere) began to take advantage of the highly variable nature of most audio signals (especially musical sources) and to cite the peak output (quite brief and rarely sustainable for long) as the amplifier power. There being no standards, imaginative approaches came to be so common that the US Federal Trade Commission intervened in the market and required all amplifier manufacturers to use an engineering based measure (root-mean square) in addition to any other value they might cite.