We report thermodynamic properties, magnetic ground state, and microscopic magnetic model of the spin-1 frustrated antiferromaget Li$_{2}$NiW$_{2}$O$_{8}$ showing successive transitions at $T_{rm N1}simeq 18$ K and $T_{rm N2}simeq 12.5$ K in zero field. Nuclear magnetic resonance and neutron diffraction reveal collinear and commensurate magnetic order with the propagation vector $mathbf k=(frac12,0,frac12)$ below $T_{rm N2}$. The ordered moment of 1.8 $mu_B$ at 1.5 K is directed along $[0.89(9),-0.10(5),-0.49(6)]$ and matches the magnetic easy axis of spin-1 Ni$^{2+}$ ions, which is determined by the scissor-like distortion of the NiO$_6$ octahedra. Incommensurate magnetic order, presumably of spin-density-wave type, is observed in the region between $T_{rm N2}$ and $T_{rm N1}$. Density-functional band-structure calculations put forward a three-dimensional spin lattice with spin-1 chains running along the $[01bar 1]$ direction and stacked on a spatially anisotropic triangular lattice in the $ab$ plane. We show that the collinear magnetic order in Li$_2$NiW$_2$O$_8$ is incompatible with the triangular lattice geometry and thus driven by a pronounced easy-axis single-ion anisotropy of Ni$^{2+}$.
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