You can play harmonics by placing a finger very lightly on the string and plucking the string. To get them to sound brighter, try playing close to the bridge.
You can learn more about harmonics by checking out a physics book, but here is a little info: When you pluck the string, it vibrates, and harmonics occur naturally at what are called "nodes" of the string. If you touch the string lightly at a node, you can emphasize that harmonic. The easiest ones to produce is at the 12th fret (the halfway point between bridge and nut) and so on, as the previous post said.
Tuning with harmonics: Using the E string as an example, you would play the 5th fret harmonic on the E, and the 7th fret harmonic on the A. it should be the same note. I have read, however, that this method is NOT as accurate as equal-tempered tuning (ie, fifth-fret/open string or octave tuning).
Tapping harmonics: Similar to fingerboard tapping, except you have to hit the harmonic dead-on and then remove your finger quickly. You can do some neat-sounding stuff with this!
Ever wish you could play that 12th fret harmonic with any note, and not just the note that the string is tuned to? For example, if you wanted to do a G fundamental harmonic on the E string, fret the G (3rd fret) with your left hand and tap the harmonic at the G on the 15th fret. This works because when you finger the 3rd fret, the 15th fret is suddenly the halfway point between nut and bridge. Now you can learn all your old songs in harmonic form.