This is a time-of-flight image, and it therefore refers to the momentum distribution of the atoms. As explained in the original Nature paper (10.1038/415039a), the distance between the peaks (in the picture, i.e., in the momentum space) is $2\hbar k$, where $k = 2\pi/\lambda$ is the wavenumber corresponding to the periodicity of the lattice with $\lambda$ being the laser wavelength.
On the top left the system is in the superfluid (Bose-Einstein condensed) phase, as indicated by the peaks. All the atoms are completely delocalized over the whole lattice, and they thus are coherent.
On the bottom rightmost picture, the peaks are gone, thus there is no phase coherence. The system is in the Mott phase, where each atom is trapped in a single site.
In between the two extremes the Superfluid/Mott phase transition takes place. There, as the depth of the lattice is increased, the atoms first become localized a tiny bit (indicated with the extra peaks appearing and gaining strength), and then the incoherent background takes over (the system forms the layered-cake structure of several Mott states with different occupancy).