In Fourier analysis, a "tone" (a sine wave) within a sound, which is (usually) higher in frequency and lower in volume than the fundamental tone, the latter of which (usually) determines the listener's perception of pitch. Overtones contribute to the perception of timbre.
Harmonics are a subset of overtones. Confusingly, after the first harmonic which is the fundamental, the second harmonic is called the first overtone, the third harmonic is called the second overtone, and so on. Most accoustic musical instruments which produce a definitive tuned pitch (i.e., not drums or noisemakers) have overtones which all fit into the harmonic series. The major exceptions are bells and chimes, which usually produce several overlapping sets of overtones, some of which are non-harmonic with respect to the fundamental. These can effect the perception of pitch of a bell or chime in odd ways. This is why synthesis methods such as ring modulation and frequency modulation are good at producing bell-like sounds, because they create a set of non-harmonic overtones that can mimic the behavior or a bell.
Non-tuned-pitch instruments such as drums produce a "spray" of overtones centered about the various vibratory modes of the drum head or surface. A group of closely clustered overtones tends to be perceived as noise. In theory, white noise has an overtone at every possible frequency.