From the beginning of time humans have been creating musical instruments which produce vibrating air at certain frequencies. All of these instruments have the same fundamental elements: an excitation source (the trigger that produces sound, e.g., plucking a string), a wave guide (somewhere for the vibrating wave to travel, e.g., down a guitar string) and finally a resonator (a cavity for the vibrating air to resonate in usually resulting in an increase in volume, e.g., the body of an acoustic guitar). .
Each instrument produces a wave, this wave has the same properties as any wave but the most tonally important being the frequency. The physical frequency of the wave determines the pitch of the sound produced once it is interpreted by our ears. This frequency is determined by how often the wave performs one complete cycle and this can differ depending on the substance the wave is traveling through. This also directly relates to the wavelength of the wave, as the longer the wavelength the lower the frequency as the wave will perform full cycles less frequently.
Many years ago a group of monks developed the first ever musical system, this was essentially a way of giving a frequency a specific place within reference to other frequencies in a hierarchical system. From then onwards we have been developing various systems in order to relate various frequencies. These systems were put into place on our musical instruments of today. Take the piano for instance, were someone to play an A(2) on a piano they would really just be striking a string to produce a vibration of frequency 110Hz and a B(2) would be 123.47Hz and so on. It is thanks to mathematicians like Pythagoras that a mathematical system was developed to be the basic link between these frequencies.
Harmonics are essentially two or more notes at specific frequencies which when combined produce one complete tone with both a fundamental (the first note) and an overtone (the harmony).