We analyze the spectral properties of a family of discrete Schrödinger operators Hμ(K), K∈T2. We partition the real line into five connected components, within which the number of eigenvalues of the operator Hμ(0) lying below and above the essential spectrum is constant. Necessary and sufficient conditions on μ are established for Hμ(0) to have exactly α eigenvalues below or above the essential spectrum, where α∈{0,1,2,3,4}. Finally, we provide sharp lower bound for the number of eigenvalues of the operator Hμ(K), K∈T2, lying outside the essential spectrum.