If a guitar is playing the A string, its fundamental frequency is 110Hz.
The string produces harmonics in addition to that fundamental, at integer multiples of the fundamental. So the second harmonic would be at 220Hz and the third harmonic would be at 330Hz.
Instruments like bells, drums, synths sometimes produce frequencies that aren’t integer multiples of the fundamental. Bells in particular are pretty “out”, meaning the relationship of their frequency components to the percieved fundamental is unclear or muddy.
In that case, the frequencies are called partials. They literally aren’t “in harmony” with the fundamental, so we can’t call them harmonics!
You can think of partials as being a superset of harmonics. It’s all the frequencies in a note, whether or not they are harmonic (integer multiple of fundamental) or inharmonic (strange frequency!).
We have a mistranslation of Helmholtz to thank for this additional piece of jargon.
Basically, an overtone is “any tone above the fundamental”. Pretty simple.
But it can get complicated because people refer to harmonics in terms of overtones.
In the case of the A guitar note, the fundamental is 110Hz. The second harmonic is 220Hz. Well, because 220Hz is the first frequency above/over the fundamental, then it’s the first overtone.
People like to make things complicated though. You’ll see daunting looking charts trying to equate partials vs. overtones. This is all unnecessary — just remember that the “over” in “overtone” means that it excludes the fundamental, so the 3rd harmonic is the 2nd overtone. The 125th harmonic is the 124th overtone.
So overtone is just
harmonic number - 1, because we don’t count the fundamental. Not tooooo difficult in the end!