There are four qualities of tone: pitch - the frequency at which the tone vibrates; duration - the length in time of the tone; volume - the amplitude of the repeating wave; and timbre. Timbre (rhymes with "amber") refers to the quality of the sound, or put more simply, how does the sound "sound." Thus we can speak of the timbre of a flute in comparison to the timbre of a clarinet, or the difference in vocal timbre of an old lady versus a child.
Recall that sound is the result of periodic waves moving through the air to your eardrum. When a sound is made, whether through vibrating a column of air (in the case of a wind instrument) or exciting a string whose ends are fixed, the object vibrates not only at its fundamental frequency - the full length of string or column, called the fundamental, but also at various other frequencies whose speeds are related proportionally to the fundamental frequency. These other "tones" are known as overtones or partials. Thus if a full string vibrates at 60 Hertz (cycles per second), segments of that string vibrate at speeds inverse to their length: each half vibrates at twice the fundamental frequency, each third at three times the fundamental frequency and so forth.
The timbre or unique quality of each musical instrument is a result of the volume of the fundamental and the quantity and volume of its overtones. Timbre differs from instrument to instrument and also within the same instrument playing in different registers. Timbre can be influenced by the strength at which the vibrations are put into motion, the noise present during the attack - or intitiation - of the sound, the speed at which that attack decays into the sustained sound, the shape or composition of the instrument or the volume of the sound.
A spectrograph presents a visual representation of timbre, by showing the volume and frequency of a sound. Comparing spectrographs demonstrates how the same pitch played by different instruments can have an entirely different spectral footprint.
The image at left above shows the spectrograph of a flute playing middle C, while the image at right is that of a clarinet playing the same pitch over a four-second span. The x-axis represents time, while the y-axis shows frequency in hertz (pitch) - our ear can hear from something like 18 Hz to 20,000 Hz. The brightness of a color indicates the volume of that frequency. Looking at the two spectrographs, we can see that the primary difference between the timbre of a flute and that of a clarinet lies in the strength of the fundamental, which is much stronger in the clarinet than in the flute. Certain overtones of the flute, namely the octave-and-a-fifth and double-octave, are stronger than those of the clarinet.
Through the years, composers have become more and more attuned to the timbres of individual instruments and their combinations and have learned to use different timbres for their expressive possibilities. Owing to folk associations, metaphorical interpretations of sound quality, and repeated use, certain timbres have become associated with particular moods, emotions or situations. These associations are historically grounded and a particular timbre can be associated with more than one mood, emotion or situation - in the Baroque, for instance, viols - essentially fretted cellos - were associated with angels, an association that is lost on the modern listener, angels having switched to the harp. That being said, through their repeated use, certain associations have been more long-lived. The assocation of the oboe or its lower-pitched cousin, the English Horn with pastoral or outdoor scenes combined with a Romantic notion that such scenes are seeped in melancholy is particularly long-lived. To cite a few examples: the Shepherd's duet in Berlioz's Symphonie Fantastique, the Shepherd's tune in Tristan and Isolde the theme from the Largo of Dvorak's New World Symphony, or the pastoral of Rossini's William Tell Overture.
The act of choosing particular instrumental colors for particular musical moments is known as orchestrating. Ravel's classic Bolero demonstrates how a melody takes on different characters depending on how it is orchestrated.