Natural harmonics

The physics of vibrating strings is a complex subject in which I have no expertise, other than twanging music out of them, so I won't pretend to know the details. I do know that when you strike a guitar string, the note that you hear is called the 'fundamental'. It's by far the loudest note created, but along with it you are also hearing 'harmonics'. These are subsidiary tones that accompany the fundamental and are responsible for making each instrument sound the way it does.

Guitar 'harmonics' are created when you lightly touch the string with your finger at specific positions and then pluck the string. This causes both sections of the string, to the right and left of the spot you're touching, to vibrate simultaneously, giving a bell-like quality to the note. It's best to quickly remove your finger as you pluck. What you are doing is damping the fundamental and only hearing the harmonics. There are three main points along the string where you can this: the fifth, seventh and twelfth frets ... directly above the fret-wire. At these points, the string is divided exactly into fourths, thirds and halves respectively. The purity of sound comes from these perfect fractions of string ringing together. If you try it anywhere else on the string, you hear a dead sound without any ring.

The video shows the technique quite well. 

Artificial harmonics

Artificial harmonics are those created on a string that's already fretted by the left hand. The right hand must do the light touch AND the pluck at the same time, AND the position of the light touch must be five, seven or (usually) twelve frets away from the fretting hand ... a lot to think about. You can arpeggiate whole chords using artificial harmonics by following the same shapes as the chords your fretting hand is forming ... 12 frets up the neck. So you have to imagine that the 12 fret is the nut, and then you have to see those shapes up there and gently touch the strings exactly above the fret-wire, all the while plucking them with your ring finger. If you have a lot of time on your hands, this is something you can practice.

Pinch harmonics

Pinch harmonics are the same as artificial harmonics, but instead of using your index finger and ring finger, you use your thumb and index in a pinching action on the string. You need to touch the string with the side of the thumb (above the appropriate point on the string) and then pluck it with the index finger just behind the thumb. It's a lot easier to on electric guitars and if you move your thumb along the string as you continue plucking, you will hear a whole lot of harmonics ring out as your thumb crosses over the nodes, which is where the string's vibrations cross over.

The videos above show these techniques much better than I can describe them in words!