Anything that changes the voltage (or current) waveform from a pure single-frequency sinusoid (in the case of AC) or a pure "flat line" (in the case of DC) is considered to be harmonic distortion.
In actual fact, there are transient conditions - events measured in the 0.001 to 0.05 second range - which cause such distortions very frequently (typically on the order of several times a second), even around the home. Such instances can be caused by switching (physically closing or opening a switch, triggering the switching action on power electronic devices such as thyristors and transistors, breakers opening/closing) or line surges (lightning strikes, starting/stopping a rotating load, step changes in loading any electrical device). The impact of such a transient is dependent on the location of the event relative to the observation point as the energy of the transient decays over the resistance of the path. This is why a lightning strike in the next county doesn't even cause your lights to flicker in your home - but one hitting your neighbor's house probably blows every fuse in the panel.
The problems associated with harmonic distortion occur when the event is no longer "transient" - which means it lasts at least for several seconds, if not minutes or hours. In this case, the multiple frequencies are carrying some amount of energy (each) - some of which adds to the normal voltage and current at line frequency, and some which counteracts it. If enough is additive, the net result is a higher voltage and/or current than expected - which leads to potential damage of the equipment using that waveform. The damage can be a result of dielectric breakdown (voltage "spikes") and/or thermal degradation (excessive current) of the windings. It may also appear as damage to the bearing surfaces, if the energy is sufficient to bridge the gap between shaft and journal.
An inexhaustive list of possibilities behind a system's harmonic distortion contains some or all of the following: switching action, operation of nonlinear devices, unbalanced phase loading (more single phase loads on one phase over another, or a motor with a damaged winding still running a process), "ringing" action caused by improper balancing or R-L-C circuitry in a specific branch, and electromagnetic interference from another branch or equipment.
There are usually limitations on how much distortion equipment is supposed to be able to withstand. One of these sources quoted fairly frequently in industrial applications is the standard IEEE 519, which details allowances for both voltage and current harmonic amplitudes. Note that the standard was originally written prior to the "electronic age" - which means there were very few switching devices related to power electronics and almost no "automatic" switching events, which in turn meant a fairly steady sinusoidal waveform for distribution purposes. It has since been revised (moving the acceptable limits upward) to account for the predominantly distorted waveforms present at all levels in today's distribution systems.