harmonics
Pluck a guitar string and you do not get just one pure tone, you get a rich blend: the main note plus fainter, higher notes stacked above it. Those higher notes, at whole-number multiples of the lowest, are the harmonics. The question it answers is: why does one string sound like a whole chord of related pitches?
The harmonics are the set of frequencies at which a system naturally makes standing waves, and they come in a simple pattern, whole-number multiples of the lowest one. If the lowest (the fundamental) is f_1, then the harmonics are f_n = n f_1 for n = 1, 2, 3, ... For a string fixed at both ends, f_n = n v / (2L), where v is the wave speed and L the length. Each harmonic is a standing-wave pattern with more nodes and antinodes than the last.
The mix of harmonics, which are strong and which are weak, is what gives each instrument its timbre, so a violin and a flute playing the same note sound different. An honest note on words: harmonics are numbered from the fundamental (the 1st harmonic is the fundamental itself), while overtones are numbered above it (the 1st overtone is the 2nd harmonic). Also, some systems (like a pipe closed at one end) produce only the odd harmonics, not every whole multiple.
A guitar string with fundamental f_1 = 200 Hz also rings at 400 Hz, 600 Hz, 800 Hz, ... (its 2nd, 3rd, 4th harmonics). Lightly touch the string at its midpoint and you silence the fundamental, leaving the pure 400 Hz second harmonic, a harmonic that guitarists use on purpose.
Harmonics are the whole-number-multiple frequencies f_n = n f_1 that a system naturally sounds.
A pipe closed at one end sounds only the odd harmonics (f_1, 3 f_1, 5 f_1, ...), so every whole multiple is not a universal rule.