The density of states $D(E)$ is the number of states at a given energy $E$. For a finite system, this is rather boring, because there are only states at discrete energy levels - that's what you're talking about. Assume a system with the energy levels $E_1$,$E_2$,$E_3$,$E_4$, then the density of states is
This expression says that there is one state at the energies $E_i$, but none elsewhere. For continuous systems, this is more interesting. The number of states can differ between energies, e.g. in Silicon: http://nanohub.org/resource_files/2011/11/12603/slides/026.01.jpg. This is particularly important because the gap in this picture between 0 and approx. 1 eV is what makes silicon a semiconductor.