Harmonics occur on the string at points that divide the string into equal parts. Lightly touch a string right above the 12th fret bar and while doing so strum the string -- you'll notice that the string sounds an octave higher, though different from just fretting the 12th. What you're doing is dampening the fundamental frequency and "odd" harmonics (3x, 5x, etc.) of open string, leaving the "even" harmonics (the predominant one being 2x the frequency of the fundamental, which is an octave higher). Notice that the 12th fret is exactly halfway the length of the string. You can do the same as the 7th fret (1/3 of the string) and the 5th fret (1/4 of the string), for example.
Now pick a note, say the 12th fret on the D string. Fret that note, and pick the string exactly halfway between that fret and the end of the string (the end on the body of the guitar of course, not the neck). You might have to experiment a little to find the exact halfway point to pick, since it's not over a handy fret marker. You'll notice a similar effect to the above harmonics. You can do this for any note. Pick another fret, find the halfway point between that fret and the end of the string, and pick the string. Same effect.
To intensify the effect, you need to lightly touch the string at this point as quickly as possible after picking (see yossarian's answer). This will bring out the harmonic more instead and keep the fundamental frequency dampened.
Just like with regular harmonics, you can also pick a point 1/3 or 1/4 of the way along the string instead of 1/2 of the way, although this is more difficult.