Since no electronic device is perfectly linear (meaning the output exactly equals the input) harmonic distortion is a fact of life in all audio components. Most audio signals have harmonics associated with them (a perfect sine wave is one notable exception), and that is what gives them their characteristic sound. An oboe sounds different from a violin mostly because of the harmonic series produced as part of their distinct sounds. The corresponding difference in the shape of their respective waveforms is easily distinguished when viewed on an oscilloscope or a computer audio editing program. Harmonic distortion is the result of a device subtly, or not so subtly, changing the shape of the waveform which alters the relative levels of various harmonics associated with that sound. The more harmonic distortion there is the more the sound will begin to take on the quality we all know and love that we call “distorted”.
In spec land you will often see the specification for THD which stands for Total Harmonic Distortion. This is a rating given to most gear for the overall percentage of harmonic distortion added to the signal passing through the device while operating at (presumably) nominal levels. There are dozens of ways to measure this spec that can skew the results so keep that in mind when comparing product literature.