Each musical note is a frequency, or sound wave. When this sound wave is shorter, it will sound like a higher pitch. Conversely, when the wave is longer, it will sound like a lower pitch. When musical notes are played, they don't send out a single frequency, but rather several at a time. These "extra" frequencies are overtones or harmonics.
Timbre (pronounced "tam-ber") is the uniqueness of a note's pitch. For instance, the A note above middle C on a piano will sound different than the A note above middle C played on a guitar, even though both of these A notes have the frequency of 440 Hz (or 440 vibrations per second). What sets the two pitches apart are the instruments' harmonics and overtones. Most instruments, including the human voice, have harmonics and overtones to them, which give each instrument its character.
While a note is played, we mostly hear the main vibration. But there are also other oscillating waves vibrating at the same time as the main note. Harmonics are specifically designated as the sound waves that are whole number multiples of the fundamental frequency. For instance, the A above middle C has a fundamental frequency of 440 Hz. Its first harmonic, then, is 880 Hz. Its second harmonic is 1320 Hz (or three times the fundamental frequency). However, there are other tones happening in between these whole integers, and these are called overtones.
The terms "overtones" and "harmonics" are often used interchangeably, and this isn't necessarily wrong. Overtones are generally defined as sound frequencies that are produced at the same time as a fundamental note, but vibrate at different rates. However, unlike harmonics, overtones can be more specifically defined as sound frequencies that don't have a whole integer relation to the fundamental note. For instance, the first harmonic for A above middle C is 880 Hz. There are more complex overtones in between these two notes, though, and any frequency vibrating between 440 Hz and 880 Hz while an A above middle C is played could be dubbed an overtone. Basically, overtones, like harmonics, are less audible frequencies. But unlike harmonics, they don't have to have a whole number relation to the fundamental note.