I like to tell people that playing harmonics on a stringed instrument is like playing jump rope. If you went to an elementary school where jump rope was popular, it is likely that you are familiar with games where two children each hold an end of a long rope and one or more children jump over the rope as it is swung around and around. In one such game the children swing the rope vigorously enough that it divides in half. Two children jump over the rope, one over each half. The jumpers have to alternate the timing of their jumps relative to one another; they are in antiphase.
Strings, like jump ropes, can be made to vibrate as a single unit, in halves, in thirds, or in any unit fraction. To isolate one of these vibration modes we place a finger on one of the nodes of the desired harmonic. A node is a place where the string seems to be standing still, like at the center of the halved jump rope. If more than one harmonic share a node location, the largest fraction takes precedent.
The above diagram is a treasure map showing the locations of natural harmonic nodes on an equal tempered fretboard with modern dot markers. The highest two pitches mapped may be too risky to use in performance on many instruments. I feel that theoretical labels would necessitate a discussion of temperament, the harmonic series, inharmonicity, and probably many other troublesome issues surrounding stringed instrument design. Rather than worry about explaining this diagram with musical theories, let's watch it dance: