There are several types of harmonics.
The most common type is the "natural" harmonic where you touch the string lightly at certain points on the string called "nodes" and then let go very quickly after you pluck, resulting in that bell-like noise. The position of the nodes along the string corresponds to divisions of the wavelength of the vibrating string, e.g. at the 12th fret, you would be dividing the string exactly in half. The other natural harmonics occur at other divisions, such as a 3rd and a quarter of the string length, (7th, 5th).
Another type is the "artificial" harmonic which can be sounded in a number of different ways. The same prinicipal of string length division is applied above in that you must pick the correct part of the string to either strike with the fleshy part of the pick hand thumb after the plectrum has hit, or fret with an index finger (in the case of fingerstyle) while you pluck with a different finger, or tap the string hard against the fretboard at the node such as Van Halen does in the intro to "Women In Love" (VH2).
Harmonics sound easier when you crank up the gain or treble and with the bridge pickup of an electric.
Here comes the science bit:
A musical note is sounded by sound waves, which you can think of as compression of the air (or medium) through which the sound is travelling.
Depending on the speed (frequency) of the wave, you hear different pitches, i.e. higher or lower.
Consider the open A string on the guitar; the fundamental frequency of the note is 440 Hz (this means wave cycles per second). When you pluck the A string, you don't hear a pure 440 Hz note, but the sound you actually hear is the fundamental note plus a bunch of other overtones related to the fundamental. These are known as harmonics. The combination of the harmonics, the relative amplitude (volume) of each of them contributes to the overall note quality and is varied by wood of the guitar, amplification, etc, equalization etc.