The explanation and understanding can be found in the physics of vibrating systems (and a little bit in basic definitions).
A harmonic series is defined by the relationship fn = n*f1m n = 1, 2, 3, ...
If f is a frequency, say 110Hz, then the harmonics are 220, 330, 440, ... etc.
A "simple" wave form or signal has a single frequency in it and is described by a trig function, sin(2 pi f t) or cos(2 pi f t). In this sequence the lowest frequency is called the fundamental. The set of all such functions can be used to describe almost any function which is quite amazing. In math we say that the set of all functions {sin(2 pi f t), cos(2 pi f t)} for a basis for a function space. Expanding any curve as an infinite series of these basic waves is the foundation of signal analysis or harmonic analysis and synthesis (the latter being the construction of a generic function by adding together sines and cosines).
In algebra the simplest function might be a straight line, having 2 parameters, a slope and intercept. So describing a line with an infinite series of sines would seem foolish. But in vibration and wave theory the simplest function is in fact a sine. The aforementioned functions are the simplest solution to the wave equation and describe a wave train of a single wavelength or frequency. In music this would be a single pitch, in optics it would be a single color (monochromatic).
Each such eave has three parameters that describe it, a frequency, an amplitude, and a phase. Frequency you know, amplitude is related to volume or power, and phase just tells you how the wave looks at t = 0 sec. Two waves with the same f and Amp but different phase are the exact same shape but shifted.
Now for the physics.
All continuous vibrating system, tight string, air column in a tube, stiff plates, etc have an infinite set of solutions to the basic wave equation characterized by the frequency (and some other parameters). Not all of these frequencies obey the harmonic sequence but the two that come very close are (1) the tight vibrating string, and (2) standing air waves in a long thin tube. These cover most of the instruments in an orchestra.
So, how do all these vibrations come into a single note? The presence of multiple harmonics in a single note plucked on a guitar for example are the result of the attack. The initial conditions that determine how the string is set in motion. The shape of a plucked string will not be a sine or cosine but can be described as an infinite series of these functions. So when you play a musical instrument including your voice you excite all these harmonics.
The harmonic relation is pretty simple and actually rare. Most real material systems, including strings, do NOT obey the simple wave equation but contain stiffness terms that are linear, but higher order. As a result vibrations in a stiff bar, or very tight short string plucked or hammered with enough force will exhibit non-harmonic over tones. I've done harmonic analysis on very long tuning forks back in college and you can easily tweak out an octave and an augmented 5th. In fact if you are a well trained musician you can hear it! And, since the lobe pattern is frequency dependent if you turn the fork you will hear different notes dominate the sound field at your ear. It's pretty cool.
I would never expect the "harmonics" of a musical instrument to truly follow the harmonic sequence due to the fact they do not obey ideal physics equations. There are some good sources that describe the corrections to the simple models. In particular anything by Fletcher and Rossing.
