[Erithon]

Here's my attempt to break this down:

An overtone is a frequency higher than and mathematically related to the fundamental. The mathematical relationship is described in something called the Harmonic Series. All of the harmonic notes on a guitar (or any stringed instrument) are overtones derived from that fundamental.

The series occurs at the octave (12th fret), the fifth above that (7th or 19th fret), then octave again (5th or 24th fret), the major third (4th, 9th, 16th, and also somewhere beyond the end of the fingerboard), the fifth, the flat seventh, the octave... once an overtone is introduced it remains in the series as it goes higher and higher. So the further from the fundamental the more complex the overtones. Why these frets create overtones is part of the math. The octave doubles the fundamental frequency so it halves the string length; the octave harmonic does so by creating a "node" at that point which allows the string to vibrate at half the previous length. Over harmonics create nodes of 1/4, 1/3, etc. The ratio dictates the frequency of the overtone and thus which note is produced.

Overtones are naturally occurring frequencies. Back in the day, to make the distance between all notes "equal" so the circle of fifths and scales would work in any key, rather than be relative to the tonic, a slightly adjusted method of tuning was introduced. We still use this tuning today and, as a result, the more distance harmonics are slightly out of tune. For example, you'll notice that the octave harmonic sounds exactly at the 12th fret, but the major third harmonic is actually found just north of the 4th fret.

Overtones also dictate the timbre or "color" of any instrument. Only a pure sine wave is a frequency without any overtones. So the strength (or lack thereof) of each overtone is very different from instrument to instrument. This is why an A note on the guitar sound different from an A note on a piano ... and thus why a guitar sounds different from a piano. Each instrument emphasizes different overtones. This is also why orchestras tune to an oboe: its harmonic spectrum has a strong fundamental and overtones that reinforce that, rather than muddy it.

Even within individual instruments there is variation, and that is what the Rosewood thread you referred to is getting at. Different guitars will sound different to one another because they emphasize or demphasize different overtones and the fundamental. The body size, bracing, and wood all factor into this. So does where you play a note. An E on the 2nd fret of the D string sounds different from that same E on the 7th fret of the A string because the length of a string impacts which overtones are amplified.