Harmonics are integers of the fundamental frequency. Please see
here. Harmonic distortion is the effect of these additional frequencies and said to alter the character of sound, with tube amps traditionally having more prominent second and other even-order harmonics and solid state more odd order harmonics. The former tend to be more musical and the latter more analytic as a BIG generalization, Of course the sonic impact also depends on other aspects of amp. design, so YMMV. Others may be able to explain the correlation on harmonics to sound characteristics better than my rudimentary generalization above. For me, setting 2nd order to "4: and 3rd order to "2" achieves a nice balance of "tube-like" musicality versus analytic "solid state" type sound,Hope that helps, but of course YMMV.
Below is an excerpt from
here that adds a bit more detail.
Triode-based tube amplifiers tend to have very little energy in the higher harmonics, so a tube amp with 1% THD exhibits mostly 2nd harmonic with very little of the other harmonics. A 2nd harmonic tone 40 dB below the fundamental (equating to 1%) is quite difficult for the human ear to detect. Early transistor amplifiers, on the other hand, were dominated by odd harmonics, primarily the 3rd and 5th, to which the human ear is significantly more sensitive. [So] engineers realized that in order to achieve comparable sound quality with an amplifier producing mostly 3rd and 5th harmonic distortion, the maximum allowable level of distortion had to be reduced by 10 – 20 dB, to .3% or even .1% THD+N.