Strictly defined, overtones are frequencies higher than the fundamental in a musical note, while harmonics are frequencies that are mathematical multiples of the fundamental. For example, 400Hz has harmonics at 800, 1.2KHz, 1.6KHz, 2KHz, etc.
More colloquially, both "harmonic" and "overtone" are often used interchangeably to refer to frequencies beyond the fundamental in a musical note.
As you probably know, very few sounds in nature are composed of one single frequency. When you play a musical instrument, it vibrates the air at a specific frequency. That frequency is called the FUNDAMENTAL frequency, and comprises the PITCH of the musical note. For example, a fundamental A is defined at 440Hz.
However, most musical instruments don't just vibrate the air at one frequency. It vibrates the air at HARMONICS of that frequency as well (440, 880, 1320, etc). The combination of the fundamental and its harmonics makes up the character of the sound, or its TIMBRE. It's the timbre of the musical note that allows you to distinguish between a trumpet's A, and a violin's A, even though they both have fundamentals at 440Hz. Instruments differ in the relative strength of the various harmonics in a musical note. For example, one instrument may have a strong second order harmonic (880Hz, from an A,) while another instrument might not have as strong a second, but a stronger third (1320), etc.
Some instruments (like cymbals, and plucked string instruments like the piano), produce frequencies that are not mathematically pure harmonics. Strictly, they would be said to have overtones, but not harmonics. However, as I said before, colloquially "harmonic" is often used to refer to both pure harmonics, and enharmonic overtones.
Is that sort of what you were looking for?
Let me know if you want some more detail, or if you want to go into combining overtones in the air to make new fundamentals, and other more esoteric stuff.