Music or noise?
Distortion is a measurement of how linear a system is, how accurately it is in reproducing the music sent to it.
Lots of distortion
There are many different types of distortion, and many different ways of measuring them. What is noticable however is that apart from pro audio monitors, you never see any specs. This is because most loudspeakers are quite aweful! If we look at the popular THD spec, while we can routinely make an amplifier with 0.001% distortion, we’re lucky if we can get a loudspeaker to produce less than 1% THD. At low frequencies this can rise to 10, 20 even 50%!
So why aren’t we crying in horror about this? It’s partly because we’re used to hearing it, and partly because loudspeaker produce mostly even order harmonics, that we find harmonious, rather than odd order, that we find objectionable.
THD
THD (total harmonic distortion) is the most widely quoted spec because it is easy to measure and gives a good figure of merit for an audio device. As it happens, we can’t hear most distortions below about 1%, but distortion is additive, so in electronics, especially in music production, we need low distortion devices as we use a lot of them in series. In a home environment, it’s not actually that important as long as the electronics are consistent.
With loudspeakers, another interesting phenomena occurs. Music is mostly harmonically based and loudspeakers produce lots of 2nd harmonic distortion, which augments rather than destroys music. In this respect THD is not as useful as a chart showing the levels of each type of harmonic distortion. Here’s one for a SoundBucket driven really quite loud ( 90dB @ 1M)

The reference is the brown flat line, and the black is the total THD, the red the nice second harmonic distortion, and the yellow the more unpleasant third harmonic.
As you can see when the distortion is high, it’s predominantly 2nd harmonic. And although it looks gross, the distortion is perhaps 10x less than some other speakers we’ve measured, which is one of the reasons why the SoundBucket sounds so engaging.
DSP and compression
The current trend is to use cheap drivers and lots of DSP to correct for frequency response issues. This can yield good results, but DSP cannot reduce distortion. The only way around this is to use expensive drivers where money has been spent on the motors and cones.
One area where DSP can make matters worse however is with compression. One way to make a loudspeaker to sound louder is to compress the music, to reduce the dynamic range, and increase the average energy of the music. DSP can do this really easily, and many manufacturers use this technique to give an edge to their speakers. It can certainly sound impressive, but it increases distortion radically, so the more you listen to it, the more it starts to sound unpleasant.
Sadly in audio, as in life, you can’t get something for nothing…