The white dots are typically placed at harmonic points on the string. The first harmonic of the open string is the octave, halfway along the string at the 12th fret. The second harmonic is one-third the string length, and that falls at the seventh fret. On some models, you'll see that fret marked with a double-dot as well, as it's a very prominent harmonic:
The woman holding the guitar here is pointing at the octave marker, but you can see the seventh fret is also double-dotted.
You'll notice the above guitar also has fewer dots; there isn't one on the third fret, and the second octave doesn't match the first. The above layout is a more harmonic-oriented one; the average player, with a little practice, can coax out a harmonic at each of the marked frets. Now, because harmonics, by their nature, are Pythagorean "just temperament", while the guitar's fretboard is designed for "equal temperament", the actual location of the harmonic usually won't be over the fret marked by the dots, so this layout isn't an exact guide but it's useful at a glance.
Most fretboards have the frets marked as you describe, and this is a "compromise" layout of sorts that can be repeated on both octaves of the fretboard, so the guitarist can play in both octaves without losing his place. This layout is more a straight-up positioning guide, but it is still useful for identifying harmonic points.
Other harmonics of the open string besides the ones in the above picture are less prominent, though a skilled player with a well-made guitar can get them to speak. Jaco Pastorius was famous for coaxing third and even second-fret harmonics from his Jazz Bass: Jaco Pastorius - Loop Jam (incorporates part of his "Portrait of Tracy" solo).