Strongly magnetic atoms are an ideal system to study many-body quantum phenomena with anisotropic interactions. Here, we report on the first observation of the superfluid to Mott-insulator transition in presence of the anisotropic and long-range dipole-dipole interaction. We demonstrate that the dipolar interaction can favor the superfluid or the Mott phase depending on the orientation of the atomic dipoles and we observe for the first time the presence of the nearest-neighbor interation between the particles. In a combined experimental and theoretical work, we show that the system is well described by an extended Bose-Hubbard Hamiltonian, which includes the onsite dipolar interaction, the nearest-neighbor interaction, and the density-induced tunneling.