This is a playing technique used on stringed instruments such as Violins, Violas, Cellos and Double Basses, Guitars and Harps.
The sound is produced by dampening the string slightly (but not shortening it fully as in normal "stopped" notes - When you press a string onto the fingerboard on a violin or guitar fully you create a "stopped note". If you only touch the string and don't depress it against the fingerboard at certain points on the string you create a harmonic).
This guy explains the process for a violinist:
In the 20th Century (and 21st) composers were interested in developing the sound of the orchestra further. One of the tricks they liked to use was the eerie, ghostly notes created by playing harmonics on stringed instruments.
Video Example:
This piece of music is originally written by Bela Bartok, and is called "Roumanian Folk Dances". It is an example of Nationalism in 20th Century music, and in this transcription by Anton Szekely he has written one of the movements entirely in artificial harmonics. The harmonics section begins at 1 minute 55 seconds. Skip to that point and listen to the harmonics.
As you are listening, you might want to see what this looks like written down:
Do you see the notes in the box? The bottom one is an ordinary, staccato F natural. The top one, however, is a diamond shaped note not coloured in. Similar to a guitar's tablature, the rhythm is indicated by the bottom (normal) note. It also tells the player where to put their first finger and stop the string. The diamond head note is where they will "float" their pinky finger to create that harmonic. Notice that it is an interval of a perfect fourth, every time.
This ensures the harmonic will sound, as it all relies on precision for the player to shorten and dampen the string at certain points, otherwise the note will not sound, or at best it will be a squeak.
(The ornaments at bar 5 onwards are mordents)
Another way in which harmonics can be written is like this:
These are harmonics that can be played a perfect 4th above open strings - in this case the A and E strings, and they sound as D and A, so the composer needs only to write the diamond head, and no indication of where to stop the string.
And one final way you might see harmonics written - these are the easiest to play, as these 4 cut the open strings (of a violin) at exactly the halfway point, and make the clearest sound.
So you might hear the difference between harmonics on violins, harps and guitars, I have linked here two further videos showing harmonics on these instruments:
Harp:
Guitar:
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