A sound is created by a vibration which changes in pitch depending on the speed of the vibration. The pitch relates to how high or low a sound is. For example, the rumbling from a truck driving down the street is a low-pitch sound, while a dog whimpering is a high-pitch sound. Pitch is measured in a unit called hertz that quantifies how frequently the vibrating object completes a vibration cycle, hence the use of the term "frequency" to measure sound pitch.
Overtones occur any time there is a tone that is higher than the fundamental tone--any time there is more than one tone coming from a sound source, there is at least one overtone. An overtone can be, but isn't always, a harmonic. Sounds without overtones tend to sound thinner than sounds with overtones. The beep you hear when you first turn on a computer may not feature overtones and sounds less complex than the welcome sound the computer makes when the operating system loads that uses overtones.
A harmonic occurs when an overtone's frequency is a whole number multiple of the fundamental tone. For example, a violin string could be producing the "A" note at 440 hertz and harmonic tones from the wood vibrating at 880 hertz and 1320 hertz. A partial overtone occurs when the overtone does match up to a harmonic frequency.
A musical instrument like a cymbal relies on overtones to make its distinct sound. The cymbal makes several tones when hit, so every tone it makes except for the lowest frequency tone is an overtone. Also, the full sound you hear from a guitar chord uses several harmonics and partials to produce a complex-sounding tone. The sound tone you perceive in both examples is a combination of the fundamental note, harmonics and partials.
Two stretched rubber bands can be used to create a harmonic note. If two girls were stretching and plucking rubber bands, they could produce different noise pitches depending on how much they stretch the rubber bands. The first girl could stretch the rubber band so it is loose and produces a pitch at 300 hertz. The second girl could stretch the rubber band righter so that it produces a pitch at 600 hertz. The second girl's rubber band is producing a harmonic pitch of the first girl's rubber band.