The statistical properties of the two-dimensional weakly compressible decaying turbulence are studied by direct numerical simulation. Particular attention is paid on the density distributions and their properties. It turned out that they depend strongly on the initial conditions of entropy and compressible component of the flow. In the non-uniform entropy case with an incompressible flow sheet structures, which are peaks and troughs in the density field, appear because of the filamentation of entropy. They do not appear in the uniform entropy case, however. As a result the power law of the density spectrum is different between the two cases. In the non-uniform entropy case the density spectrum scales as k-1, where k is the wavenumber, since density behaves as a passive scalar. In the uniform entropy case, on the other hand, it scales as k-5 as predicted for pressure in the two-dimensional incompressible turbulence since density fluctuation is approximately proportional to pressure fluctuation. When the initial velocity field has a compressible component, weak shocklets appear even for the low Mach number M0 = 0.1. The density spectrum scales as k-3 regardless of the initial conditions of entropy.