The lowest is called the fundamental and that is the one we usually identify as the pitch of the note. Above the fundamental frequency are the overtones . Overtones can be harmonic or inharmonic . If they are harmonic then they are some integral multiple of the fundamental frequency, ie f, 2f, 3f, 4f, 5f etc. Sometimes that is called the harmonic series , or just the harmonics .
Harmonic overtones tend to blend well whilst inharmonic overtones give an edge to the tone.
The timbre of a sound will depend on the mix of overtone frequencies. If the overtones are all exact multiples of the fundamental, they will complete their wave patterns at the same time as the fundamental and hence reinforce the sound. You would probably say that the sound is coherent, full and easily recognised in pitch. Some overtones will be more prominent than others though and that will affect the timbre. Also the overtones may be slightly sharp due to the physical properties of the instrument making the tone.