A harmonic is basically a flutelike tone that comes from the guitar by touching the string at or near the node. There are 2 kinds of harmonics used in guitar playing, Natural and Artificial. This lesson will attempt to cover both in a simple manner.  

Natural Harmonics 

Lets cut to the chase and talk about Natural Harmonics a bit. Put your index finger above the 12th fret on the bottom E string. (actually, any string will do). Do not press down or add pressure, just set it on the fret. As you pick the note, pull your index finger off. 

E||--12--------||
B||------------||
G||------------||
D||------------||
A||------------||
E||------------||

You should hear a nice flutelike sound. If you don't, your timing was off. The hardest part about playing the Natural Harmonic is timing. If you pull off too soon, it'll sound normal. If you pull off too late, it'll sound muted. (if you do not know what a fret is, they are the ridges along the neck; not the spaces between where notes are usually played) Once you've done that, play all the strings on the 12th fret. 

E||--12----------------------|--------------||
B||--------12----------------|--------------||
G||--------------12----------|--------------||
D||--------------------12----|--------------||
A||--------------------------|--12----------||
E||--------------------------|--------12----||

Cool, eh? Now try these on the 7th and 5th frets respectively. If this is too easy for you, then try the 3rd fret. You really need to get the timing just right on the 3rd . Try putting your finger slightly ahead of the 3rd fret..  

When you play a single note on your guitar, you hear what is know as the Fundamental. This is the most dominant frequency. But there are also many other frequencies and fainter notes within that note, it just that the Fundamental frequency is most apparent to our ears. When we play a Natural Harmonic, we absorb the Fundamental with our finger (along with other frequencies, depending on which fret we are on) The tone of the harmonic decreases exponentially. Which means the higher the fret., the lower the tone is of the harmonic. This is because your finger absorbs much more of the energy of the harmonic. 

Artificial Harmonics 

Unlike Natural Harmonics, all the action is done with the right hand. First, put your pick down and just try 'plucking' a note with your index finger and thumb. When you play an Artificial Harmonic, you're basically plucking the note, except there is a thin piece of plastic protruding from your thumb and index finger, which is the guitar pick. Just imagine you are going to pluck and pick the note at the same time. 
 

Now grab your guitar pick tightly with your index finger and thumb. Normally you want to grip your pick low. To play Artificial Harmonics best, grip your pick even lower so that your index finger and thumb are almost toughing the string. (try using a dime or a penny for better results) Now pick the note, as you pull off the note slightly pluck it with your thumb. You should get a nice clean tone. This is not easy at first, but keep trying. Try this little riff. (The harmonics are in parentheses) 

E||----------------------|------------||
B||----------------------|------------||
G||----------------------|------------||
D||------------7----9----|--(9b)-(7)--||
A||--7----9--------------|------------||
E||----------------------|------------||

That's about all I have for you on harmonics. Hope this helps. If not, ask and I'll try to help.