Just about any sound maker or instrument produces a fundamental tone and higher overtones. It is the type of overtone produced that gives an instrument its "timbre" or "acoustic color", so that you can tell a clarinet from an oboe for instance. There are complex mathematical formulas to describe and define types of overtones. For our purpose, we define overtone instruments as those wherein the overtone is very noticeable, almost as strong as the fundamental.
A (modern) musical instrument usually is made to produce a clear and strong fundamental tone with little audible overtones just to give the instrument its unique timbre. This purity of tones allow these instruments to be played in groups as orchestras or bands without unpredictable overtones interfering with the music and to play perfect musical intervals. In contrast overtone instruments have obvious overtones that may or may not be harmonious (in harmonic intervals) to the fundamental. That is the reason why two different overtone instruments, like Tibetan singing bowls for example, even having the same fundamental note can sound quite different when they have different overtones.
The mystics considered these overtones, these notes that seemed to come from the ether itself, to be magical and powerful. The instruments that produced them were held in high regard for they were a portal to the other world.
In a more scientific approach the term overtone is used to refer to any resonant frequency above the fundamental frequency - a partial wave. When a resonant system such as a hand hammered Tibetan singing bowl, a Burma gong or a monochord is excited, a number of overtones may be produced along with the fundamental tone. Thus you will hear a fundamental tone plus overtones at the same time when you play these instruments. This is especially startling when you are playing an ancient string instrument like the monochord or Sound Bed and hear notes to which none of the strings are tuned.
Whereas stringed instruments typically produce harmonious partials, Tibetan singing bowls tend to produce inharmonious partials while the Burma gongs will produce both.
Whether you tend to the mystic or to the scientific, the unworldly sounds of these overtone instruments can enable you to free the mind and reach a different level of awareness, thus reducing stress and increasing relaxation and self awareness.