I decided to write about this, because I keep seeing the terms used inaccurately and confusingly, and I hope I can shed some light on the topic.
Simply put, harmonics are multiples of a given frequency. So if I have a frequency of 200 Hz, then it could have harmonics of 400 Hz, 600 Hz, 800 Hz, 1000 Hz etc. Where the confusion comes in is the naming of these. In this example the 200 Hz frequency is called the first harmonic or the fundamental frequency. 400 Hz is then the second harmonic, or first overtone. 600 Hz is the third harmonic, or second overtone. Why is this important? Because I see people confusing the numbering of harmonics and overtones, making things off by one. But is it such a big deal? Well, in some cases it is, because we (generally) perceive even harmonics as pleasant and odd harmonics as unpleasant. Of course this is not the full truth, and odd harmonics are important too, but they give sound a metallic, harsh tone. For example waveforms such as sawtooth are composed almost exclusively of odd harmonics, and they're not exactly pleasant to listen to. Alternatively some people enjoy vacuum tubes in their audio gear, because they tend to produce distortion dominant in the second harmonic, often described as warmth or smoothness.
And that's it in a nutshell. If anyone finds this helpful, please let me know.