At the fundamental frequency, the whole string bows back and forth in phase, and the only stationary points are at the fixed ends. At the second harmonic, the frequency is twice the fundamental, and the string vibrates in two sections with a stationary point at the very center. The two halves vibrate out of phase. At the third harmonic, the frequency is three times the fundamental, and there are two stationary points that divide the string into thirds. Adjacent thirds of the string vibrate out of phase with one another, so that the outer thirds are in phase with each other and out of phase with the central third.
Higher harmonics divide the string in fourths, fifths, and so forth.
You may have seen pictures of this in references on physics or music theory. The exception holds only for the middle node of even harmonics, which is always precisely at the midpoint of the string over the 12th fret. One way to find the nodes would be to measure the active length of the string with a meter stick, divide the length by the desired harmonic, and take multiples of this distance to each successive node. Well, that is almost right, and actually the approximation is pretty good for the 2nd, 3rd, and 4th harmonics. We can find improved locations for the 5th harmonic, though, and in principle we can find the location relative to the frets for any node of any harmonic we might want.