When you amplify a signal, any signal, the circuit you use will often have some non-linearities. These can lead to additional frequencies being produced that did not exist in the original signal. We call these "extra" signals harmonic distortion.
Every instrument has a harmonic series; there is the fundamental note and then a bunch of other related frequencies. Together they determine the timbre of the instrument - what makes it a cello, a flute, a clarinet or a piano? The harmonics largely determine this. So it goes without saying perhaps that we generally do not want to change these harmonics much when we reproduce the recorded sound of an instrument.
There isn't just one harmonic, there is an entire series consisting of N*X harmonics where N is your input and X is the series of real numbers counting by 1s. The limit is determined by both the transfer curve as well as the capabilities of the circuit in question. If the 5th harmonic is too high a frequency for the circuit to reproduce, for example, it may not get to the output. In theory if your circuit had no limits you could have harmonics at 2x, 3x, 4x 5x 6x 7x 8x and more. I made the chart here up to the tenth, but it can continue if the design is bad enough and the circuit is capable. One criticism I have heard leveled at solid state circuits is their ability to create higher order THD products than tubes. Another possibility that was suggested is because transistors tend to be used in differential pairs, they generate predominantly odd harmonics (push-pull cancels even harmonics) which are generally less pleasant. (has anyone tested this or seen an interesting paper on it?)
I mention this because in both HiFi and musical instrument amplification a lot is made of the harmonics the devices may create or add. Without context, it's not as useful to make generalizations about odd vs even or high vs low. Low and even is good, sure. 2nd harmonic is the octave. But after that nothing is really simple, except the other octaves. Someone claiming that even harmonics are all good has probably never thought about the 6th harmonic.
Here is a full set of harmonics for A440 (aka A4), up to the 10th harmonic & their various musical relationships to the fundamental and each other. The message here is, "it's complicated". You don't just get other notes, they may be sharp or flat, and they may do some awful sounding things in combination with each other as well as the tonic.
You can see the frequency "compression" happening here; how we hear is driving this. We hear a doubling of F as an octave - any doubling. 20Hz to 40Hz or 5,000Hz to 10,000Hz are all the same interval to our ears. Since our fundamental is 440Hz, we are going up in 440Hz steps which become smaller and smaller in the scheme of things as we go.
Harmonic distortion is commonly quoted as "total harmonic distortion" or THD and is given as a percentage of the output of the device or circuit. If your circuit delivers 10 watts at 5% THD, you have 10W total output, and .5W of that is various harmonic distortion components.
A THD meter is basically a precision voltmeter, a precision audio oscillator and a precision hi-Q (notch) filter all in a box. You generate a sine wave, say 1kHz and inject it into the device under test. You typically measure at the rated full undistorted output. Then you tune the notch filter to eliminate 1kHz in the device's output and finally you measure the remaining voltage. Anything left is THD because you have removed the original signal. Reference that level to the unfiltered output and you have your THD figure.
Spectrum analyzers can show you the specific harmonics being generated and their quantities. Some, like 2nd order, are fairly pleasing sounding and are sometimes said to "thicken" your sound. Others, like 5th (and above) generally become increasingly strident and unpleasant. Play 8, 9 & 10 together on a piano for example. This is all bad enough, now imagine you are playing an Am chord and you make this chart for all 3 notes and have to work out all of the interactions. Before we start we can already see our 4-5-6 triad for the A note is going to be dissonant (putting a major 3rd over a flat 3rd in our Am chord). Imagine this kind of interaction on top of a whole band. It gets horrible in a hurry, which is why we generally strive to have very low percentages of THD in our gear.
Try this out: you can add different sine waves together with sliders. You have to figure out which are octaves etc. But still fun to see what you can wind up with: https://physics.weber.edu/schroeder/software/WaveBuilder.html
Intermodulation Distortion is a similar idea - you are getting signals out that you did not put in. However the signals have a different origin. Here we have what are called sum & difference frequencies. When we measure THD, we use a single frequency. IMD is measured by injecting 2 frequencies at once. It has a pretty clear formal definition so rather than muck it up, I have copied it here (with it's source)
The formal info is here: https://www.digitizationguidelines.gov/term.php?term=intermodulationdistortionaudio#
"Definition:
IMD results when two or more signals of different frequencies are mixed together and additional signals are formed at frequencies that are not, in general, at harmonic frequencies (integer multiples) of either source signal. Intermodulation causes spurious emissions that can create minor to severe interference to other operations on the signal. Although IMD may affect many types of signal data, this definition focuses on sound. Intermodulation should not be confused with general harmonic distortion (which does have widespread use in audio effects processing; see THD). Intermodulation specifically creates non-harmonic tones due to undesired mixing of near frequencies.
There are various ways to measure IMD. In the United States, the SMPTE (Society of Motion Picture and Television Engineers) standard is often used. It specifies a two-sinewave test signal consisting of a low-frequency, high-amplitude tone linearly combined with a high-frequency sinewave at 1/4 the amplitude of the low-frequency tone, specifically tones at 60 Hz and 7 kHz. When a non-linear device is subjected to a two-tone test signal, intermodulation products appear as sidebands around the high-frequency tone. The percentage intermodulation distortion is defined as the percentage of amplitude modulation, represented by the second and third order pair of sidebands, of the high-frequency signal. Second order sidebands around the high frequency tone are spaced at a frequency equal to the low-frequency tone (FH ± FL). Third order sidebands are spaced at twice the low-frequency tone or FH ± 2FL. (FH and FL correspond to the high frequency and low-frequency tones, respectively. There is also a European measurement standard, specified by ITU-R."
Of course you can (will...) have IMD & THD in the same circuit. Non-linearity causes both.
Again, starting with 2 simple signals in the formal definition above, we can wind up with a range of signals that were never part of the input signal. And this is with a super simple example, 60Hz and 7KHz. Now imagine a guitar, or a grand piano. Or an orchestra. Extremely complex signals with huge ranges of frequencies, all generating THD and possibly IMD as well.
IMD is rumored to be one of the things guitarists enjoy from distorting guitar amps. There are lots of theories but I have yet to see an analysis. (have you?) Are our good grindy noises partially attributable to 120Hz IMD artifacts from the power supply (for example)?
While injecting in one or 2 frequencies is required to analyze a device, in reality we don't listen to sine waves. We are listening to a a real instrument that already has a series of harmonics which changes over time. Very often we're listening to a recording of a band or orchestra, a singer or chorus and just imagine all of these signals and their harmonics are getting acted on by additional harmonic and intermodulation distortion.
Adjacent: This is one big driver behind the use of negative feedback in circuits; we're adding a reversed copy of the distortion that is happening later in the circuit to cancel some of it out up front. We don't have to know exactly what will go "wrong", we are dynamically adding pre-emphasis to help counter it later in the circuit*. Crazy stuff, but it works; thank the phone company. (they invented & patented most of this stuff).
* we talk about signal flow to explain things but in reality it's all happening effectively at once.
About Musical intervals related to the harmonic series:
The chart I based mine off of: