A non-scientific answer : each sound has a basic frequency. This is called the first harmonic. Why it's a harmonic and not a basic fundamental, I don't know ! Other sounds emanate from the sound source. Those that are twice, thrice, etc. the frequency are called harmonics or overtones, the same thing.
Partials, or inharmonic overtones also exist within a sound. Rather than being 'octave copies' of the basic frequency, they are 5ths, etc. As a string is split into fractions (on gtr, for example, touching a string at fractions along its length) it produces the different overtones. As in - 1/2 way - 2nd harmonic. 1/3 way - fifth. 1/4 way - 3rd harmonic. 1/4 way - maj. 3rd.It is entirely possible, further along a guitar string, to play an octave scale, albeit not spot on, frequency wise, starting around the 2 1/4 fret going to the 1st fret.All partials, or inharmonic overtones which are found within the basic open string note, although rather in the background, aurally
A lot of these sounds can be heard when a note is played on an instrument. The proportion of each varies with each instrument. This gives each its own timbre (tone). Bells have maybe the most partials, which makes them sound like they are 'more out of tune' than, say, a flute, which has fewer.
Distortion produced by overdrive, on guitars, enhance harmonics, or overtones. This becomes a problem when whole chords are played, as each component note then has its partials magnified. All the sounds blend into noise. By using only a I and its V, we're left with overtones which sound good together, without clashing. Thus the ubiquitous 5th chord, beloved of a lot of guitarists.
Scientific answers, with graphic explanation, will, I hope, follow.
A simple straight answer is that including the fundamental, all are harmonics :440 Hz, 880, 1760, etc.(on A, say). Overtones do not include the basic. Overtones = 880, 1760, etc. So the 1st harmonic is 440, whereas the 1st overtone is 880: 1st overtone =2nd harmonic.I am quoting here - there are other quotes which say that 1st harmonic =1st overtone (as 880 Hz in the example).
EDIT: The overtones are called 'upper partials'- the fundamental note being a partial, not an upper partial.'Upper partial' is a synonym for 'harmonic', which is not quite correct, since though all harmonics (except the fundamental) are 'upper partials', not all upper partials are harmonics.Going back to the bell mentioned earlier, it has upper partials which do not correspond to the harmonic series.