The wattage is the most important sales argument for most amplifiers and speakers. It is often misunderstood and equated with volume (and the theory that more is better - like megapixels with cameras). Bigger numbers always seem to cost more, and numbers are easy to sell - A has more than B, so A must be better.
Volume is a function of SPL (sound pressure level), measured in decibels (dB). Speakers are typically specified with the maximum wattage they can handle and the sound pressure level, e.g. "87dB @ 1W/1Meter". In other words, the 87dB loudspeaker produces 87 decibels with only 1 watt of amplifier power at a distance of 1 meter.
Now consider that double the wattage is required to achieve a 3dB increase in volume level. So a 100 watt amplifier will deliver a 3dB higher volume level than a 50 watt amplifier. However, 3dB is not as noticeable as one would expect. To actually achieve a doubling of the perceived volume, an increase of 10dB is necessary.
Let's take the 87dB loudspeaker as an example first and double the amplifier power:
1 Watt = 87dB
2 Watt = 90dB (Doubling the power to achieve an increase of 3dB)
4 Watt = 93dB
8 Watt= 96dB
16 Watt = 99dB (about twice as loud as on a 1 Watt amplifier)
32 Watt = 101dB
64 Watt = 104dB
128 Watt = 107dB
256 Watt = 110dB
512 Watt= 113dB
Here you quickly realize that a higher volume not only has something to do with a more powerful amplifier, but that the loudspeaker sensitivity is also a variable.
If you have a 97db @ 1W/1Meter sensitive loudspeaker, you get the following result:
1 Watt = 97db
2 Watt = 100dB
...
16 Watt = 109dB
With the "87dB" loudspeaker, you would need about a 250 watt amplifier per channel to achieve the same volume as a 16 watt amplifier with the "97dB" loudspeaker.
Of course, it is important to keep in mind how high a loudspeaker (manufacturer's specifications) can be amplified without causing damage.
Therefore it also depends a lot on the sensitivity of the connected loudspeakers and especially on what I want to achieve/sonicate with the system. Basically, a powerful amplifier alone does not really say anything about the volume to be achieved, it's the combination that does it. Nevertheless, it can of course be assumed that an amplifier with a higher WATT specification can also generate a higher output if you compare this with a constant speaker.
But you should not be guided by crazy PMPO specifications of some products. To know what an amplifier can do and what a loudspeaker can tolerate, the RMS value should always be used. But this is a different subject, which I will discuss separately in another article.
Ps.: You will never be able to make a statement about sound quality with the WATT specification!