When you play a harmonic at the twelveth fret, what are you doing? You are cutting the string in half. It is vibrating on both sides, thereby getting the cool effect. If you put your finger on the seventh fret for example, the string vibrates again on both sides, but this time it produces the note of the string on that fret an octave higher. The reason being that the string has been shortened by half and vibrates on both sides of your finger to the same extent. It is interesting, because it has to do with phyics. Pythogoreas discovered that if you cut a string in half on any instrument, it produces an octave. Therefore, at the twelfth fret of a guitar it is an octave. The harmonic is produced by cutting the string in half using a 'node' and it vibrates on both sides causing a stereophonic effect. If you understand this, it is also easier to understand how harmonics are produced on the guitar in general.