Under investigation in this paper is a discrete (2+1)-dimensional Ablowitz-Ladik equation, which has certain applications in nonlinear optics and Bose-Einstein condensation. Employing the Hirota method and symbolic computation. we obtain the bright/dark one-, two-, three-and N-soliton solutions. Asymptotic analysis indicates that the interactions between the bright/dark two solitons are elastic. Amplitudes and velocities of the bright and dark solitons increase with the value of the coupling strength increasing. Head-on and overtaking interactions between the bright two solitons as well as the bound state two solitons are depicted. Overtaking interaction between the dark two solitons are also plotted. The increasing value of the coupling strength can lead the increasing amplitudes and velocities of the bright/dark two solitons. (C) 2017 Published by Elsevier B.V.