  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 $[01\bar 1]$ 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$^{2+}$.
