Single crystals of ${\mathrm{Yb}}^{3+}$-doped $\mathrm{Na}\mathrm{Gd}{\left(\mathrm{W}{\mathrm{O}}_4\right)}_2$ with up to $20\mathrm{mol}\%$ ytterbium content have been grown by the Czochralski technique in air or in ${\mathrm{N}}_2+{\mathrm{O}}_2$ atmosphere and cooled to room temperature at different rates $(4–250°\mathrm{C}∕\mathrm{h})$. Only the noncentrosymmetric tetragonal space group $I\overline{4}$ accounts for all reflections observed in the single crystal x-ray diffraction analysis. The distortion of this symmetry with respect to the centrosymmetric tetragonal space group $I{4}_1∕a$ is much lower for crystals cooled at a fast rate. ${\mathrm{Na}}^{+}$, ${\mathrm{Gd}}^{3+}$, and ${\mathrm{Yb}}^{3+}$ ions share the two nonequivalent $2b$ and $2d$ sites of the $I\overline{4}$ structure, but ${\mathrm{Yb}}^{3+}$ (and ${\mathrm{Gd}}^{3+}$) ions are found preferentially in the $2b$ site. Optical spectroscopy at low $\left(5\mathrm{K}\right)$ temperature provides additional evidence of the existence of these two sites contributing to the line broadening. The comparison with the ${}^{2}F_{7∕2}\left(n\right)$ and ${}^{2}F_{5∕2}\left(n^{\prime }\right)$ Stark energy levels calculated using the crystallographic Yb-O bond distances allows to correlate the experimental optical bands with the $2b$ and $2d$ sites. As a novel uniaxial laser host for ${\mathrm{Yb}}^{3+}$, $\mathrm{Na}\mathrm{Gd}{\left(\mathrm{W}{\mathrm{O}}_4\right)}_2$ is characterized also with respect to its transparency, band-edge, refractive indices, and main optical phonons. Continuous-wave ${\mathrm{Yb}}^{3+}$-laser operation is studied at room temperature both under Ti:sapphire and diode laser pumping. A maximum slope efficiency of 77% with respect to the absorbed power is achieved for the $\pi$ polarization by Ti:sapphire laser pumping in a three-mirror cavity with Brewster geometry. The emission is tunable in the $1014–1079\mathrm{nm}$ spectral range with an intracavity Lyot filter. Passive mode locking of this laser produces $120\mathrm{fs}$ long pulses at $1037.5\mathrm{nm}$ with an average power of $360\mathrm{mW}$ at $\approx 97\mathrm{MHz}$ repetition rate. Using uncoated samples of $\mathrm{Yb}:\mathrm{Na}\mathrm{Gd}{\left(\mathrm{W}{\mathrm{O}}_4\right)}_2$ at normal incidence in simple two-mirror cavities, output powers as high as $1.45\mathrm{W}$ and slope efficiencies as high as 51% are achieved with different diode laser pump sources.