The most natural example of sound production in the nature is of a vibrating single string. We all know that a string tightly stretched across 2 ends produces a sound on plucking.
"1st Harmonic" or "Shadja" or"Fundamental Tone"
When a string is made to vibrate by plucking, to start with it vibrates in it's full length; and the makes a certain sound. This is called as "Shadja" or the "Fundamental Tone",or "1st Harmonic" . For the purpose of understanding, we shall take it's frequency as 100 hz.
"2nd Harmonic" or "Tara Shadja"
Immediately thereafter, as the energy put in the string for plucking reduces, the string starts vibrating in 2 parts. This produces a sound of 200 hz called as the 2nd Harmonic. This is "Tara Shadja" as we know.
"3rd Harmonic" or "Pancham"
Immediately thereafter, as the energy put in the string for plucking reduces further, the string starts vibrating in 3 parts. This produces a sound of 300 hz called as 3rd Harmonic. This is however perceived by our brain as of 150 hz (our brain has the spectrum of perception of 100 hz to 200 hz) or "Pancham" (5th) as we know.
"4th Harmonic" or "Ati-Tara Shadja"
Immediately thereafter, as the energy put in the string for plucking reduces further, the string starts vibrating in 4 parts. This produces a sound of 400 hz called as 4th Harmonic. This is perceived by our brain as of 200 hz (our brain has the spectrum of perception of 100 hz to 200 hz) or Tara Shadja again.
5. "5th Harmonic" or "Gandhar" :
Immediately thereafter, as the energy put in the string for plucking reduces further, the string starts vibrating in 5 parts. This produces a sound of 500 hz called as 5th Harmonic. This is perceived by our brain as equivalent of 250 hz or 125 hz or Gandhar.
After this stage, the energy put in the string for plucking reduces so much that further harmonics (6th, 7th, 8th and so on) are barely heard.
Harmonics :
The Sounds created which are in multiple proportions to the frequency of the Fundamental or Shadja (100,200,300,400,500 as above), are called as "Harmonics", because they are in "harmony" with the Shadja. They sound very pleasant to the human ear. We all have experienced that Pancham (3rd harmonic) and Gandhar (5th harmonic) are most pleasant. Tanpura creates an atmosphere of dominant Shadja-Gandha-Pancham.
Partials :
When a sound source produces 'other than harmonic' frequencies, they are termed as "inharmonics" or "partials". These are not in harmony with Shadja and are unpleasant to the human ear.
Timbre (Harmonic Profile) :
When the source object of sound (string in this case) vibrates again and again, different harmonics/partials generated mix, and this mixture is interpreted by human ear as having a 'particular' quality. This Harmonic profile of the source is termed as "Timbre" or "Nadaguna".
The Timbre of every sound source is 'typical and different' from each other. That is how we recognize sounds of different instruments and even sounds of people, by their 'typical and different' Timbres.
The Harmonics coming out of string instruments which have long strings like the Violin and Sarangi have a Harmonic profile similar to that of human voice and hence are considered as best for accompaniment to human singing.