Harmonics are a beautiful addition to guitar playing as they have a lovely sound that complements a piece of music. There are a number of types of harmonics and I will illustrate them here.
Natural harmonic - This is played by lightly resting the fretting finger directly over the fret and playing the string. The finger is then almost immediatley removed from the string. These mainly occur at the 5th, 7th and 12th frets but the guitar is littered with them. Some of them aren't even on the fretboard! Try playing these harmonics over the 5th, 7th and 12th and then when you're confident, try finding these harmonics somewhere around the pickups or soundhole. Some of my favourite harmonics to include into music are found around the 2nd, 3rd and 4th frets too!
Artificial harmonic - play the note as normal but immediatley try to catch the string with the very edge of your thumb. Once you get your timing right and your picking angle spot-on you will be able to hit them every time. Hint: try to hit into the string rather than away from it. I find that a slight twist of the wrist at the moment of contact brings the thumb into good contact with the string.
Palmed harmonic - The note is played as normal but the palm has to catch the string ever-so-lightly a split second after picking.
Pick-touch harmonic - The note is played as normal and then you touch the string really lightly with the edge of your pick.
Stroke harmonic - This is the same as the pick-touch only you use your finger instead of your plectrum. There is a certain way of holding the pick for this method that kind of allows you to lightly 'stroke' the string at the same time as you pick it.