Overtones can be best described as a series of harmonic information. They often accompany the fundamental pitch of a note, creating a tasty complexity or timbre. They usually are multiples of the fundamental pitch. For instance, if you were to play your open A string, as the note decays, overtones will be present that give the singing string it’s tone or color. The A string is vibrating at 110Hz. The overtones present may be at octave points like 220Hz, again at 440Hz, 880Hz, etc. Or more interestingly, perhaps the overtone series contains more musical tones like the Major 3rd or Perfect 5th.
Now, instead of a composite timbre of just the fundamental note you played (A) and it’s higher unison octaves, you have the complexity and rich timbre of chordal harmony. This gives the string its own sound and distinctive quality. The instrument has this feature built into it. We may perceive the fundamental note we played as the pitch due to its dominant nature with respect to frequency, but it’s little harmonic buddies are there, in smaller amplitude, to lend some tone! This natural characteristic of the guitar is one of the reasons I love using electronics with it. The gear, whether it be pedals, amps, plug-ins, or whatever, adds its own overtones to the equation creating vast permutations and variation to this already “harmonic” sound. This is especially true with tubes, which are known for their rich, warm, and musical tone.