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Harmonics are whole number multiples of the basic pitch of a note. Many timbres are made up entirely of harmonics of various strengths.
Sounds of a flute, recorder, or ocarina can be simulated with just two or three harmonics. Other instruments will need a fair number, especially, string instruments have many harmonics.
A string player can selectively sound harmonics of a note by playing a note, and lightly touching on the string in various places.
It's also possible to train to hear harmonics.
One can also try singing the partials to a drone, and this is a lot of fun, and anyone can do it. Just play a drone, e.g. on a cello, then try to sing an octave above, an octave plus a fifth and so on - one can use a clip such as this one to give one guide tones to find the notes easily.
Here is the 'cello drone without the guide tones:
Even if you can't hear the harmonics in the timbre as such, you will very likely find you can sing them to the timbre as a drone, perhaps using a guide tone to find them the first few times. This is also a great way to learn to sing pure (just intonation) intervals.
One field where the recognition of partials in a timbre has been developed to a high degree is the craft of bell making. The more general term for a constituent frequency of a timbre is "partial", and bells have inharmonic partials - frequencies that aren't constrained to simple multiples or near multiples of the basic frequency.
The 17th century carillioneur (and recorder player) Jacob Van Eyck pioneered modern methods of tuning bells. This is done by ear, in combination with measuring instruments.
Interestingly, bell tuners also use ratios to target the partials. They make some use of the so called sub-harmonic series (the inverses of the harmonics) for the first few partials of a bell, favouring the ratios 1/12, 1/6 1/5 1/4 1/3.
http://www.oakcroft13.fsnet.co.uk/lehr.htm
This gives the characteristic minor chord type sound of a church bell.
This is because the physics of the bell makes it hard to target a major third. Major third bells do exist. Can be by serendipity as for this bell: but one can also nowadays make major third bells to order.
Another pattern of partials that concern bell founders are doublets - closely spaced pairs of partials caused by a small deviation of the bell from perfect symmetry.
Here is a fascinating web site about bell partials, which also has an excellent user - friendly program, designed for bells,that one can also use to find the partials in any instrument.
http://www.oakcroft13.fsnet.co.uk/index.htm
The numbers such as 9/8, 6/5 and 5/4 that one sees so often in definitions of scales often come straight from the harmonic series. So for instance, you get the 5/4 - major third, from the ratio of the fifth and fourth multiple of the asic note. The 6/5 is a minor third, 3/2 is a major fifth and 9'8 is a whole tone.
10/9 is another form of the whole tone. If you play harmonics 8 9 10 12, you get four of the five notes of the just intonation pentatonic scale. 8/8 = 1/1, 9/8, 10/8 = 5/4, and 12/8 = 3/2.
It is perfectly possible to play this fragment scale on the harmonics of a string instrument (and on other instruemnts that can be played in this way).
The missing note is the major sixth which is a third below the octave, so is at a ratio of 8/5 (i.e. 4/5 of 2/1)
The missing note is the major sixth which is a minor third below the octave, so is at a ratio of 5/3 (i.e. 5/6 of 2/1). Note that this is alsoa 10/9 whole tone above the 3/2.
You won't find this one in the same fragment of the harmonic series as it would be at a ratio of 40/3 to the fundamental.
However, if you multiply all the numbers by 3, you can then find the entire pentatonic scale in the harmonic series as
(8 9 10 12 40/3) times 3 = 24 27 30 36 40
These harmonics are a bit high even for a stringed instrument (can a skilled player play these? anyone know).
Some instruments use notes from the harmonic series when played normally, notably the natural trumpet of course. It is also possible for a flute player to play the harmonic series on a hosepipe by blowing across the end, playing it as an end blown flute.
There is quite a degree of interest also in scales based on inharmonic partials - i.e. using the constituent frequencies of a timbre such as a bell sound to make a scale.
There is also some speculation that the Indonesian gamelan scales may have originally been inspired partly in this way. (Ok to say this?)