We present the first full catalog and science results for the $Nuclear\; Spectroscopic\; Telescope\; Array$ ($Nu$STAR) serendipitous survey. The catalog incorporates data taken during the first 40 months of $Nu$STAR operation, which provide ≈20 Ms of effective exposure time over 331 fields, with an areal coverage of 13 deg$\mathrm{<mrow\; data-mjx-texclass="ORD">2</mrow>}$, and 497 sources detected in total over the 3-24 keV energy range. There are 276 sources with spectroscopic redshifts and classifications, largely resulting from our extensive campaign of ground-based spectroscopic follow-up. We characterize the overall sample in terms of the X-ray, optical, and infrared source properties. The sample is primarily composed of active galactic nuclei (AGNs), detected over a large range in redshift from $z$ = 0.002 to 3.4 (median of $⟨z⟩=0.56$) = 0.56), but also includes 16 spectroscopically confirmed Galactic sources. There is a large range in X-ray flux, from log(f/erg s$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ cm$\mathrm{<mrow\; data-mjx-texclass="ORD">-2</mrow>}$) ≈ -14 to -11, and in rest-frame 10-40 keV luminosity, from (L/erg s$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$) ≈ 39 to 46, with a median of 44.1. Approximately 79% of the $Nu$STAR sources have lower-energy (<10 keV) X-ray counterparts from $XMM-Newton,\; Chandra,$ and $Swift$ XRT. The mid-infrared (MIR) analysis, using $WISE$ all-sky survey data, shows that MIR AGN color selections miss a large fraction of the $Nu$STAR-selected AGN population, from ≈15% at the highest luminosities (LX > 10$\mathrm{<mrow\; data-mjx-texclass="ORD">44</mrow>}$ erg s$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$) to ≈80% at the lowest luminosities (LX < 10$\mathrm{<mrow\; data-mjx-texclass="ORD">43</mrow>}$ erg s$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$). Our optical spectroscopic analysis finds that the observed fraction of optically obscured AGNs (i.e., the type 2 fraction) is F = 53$\mathrm{<mrow\; data-mjx-texclass="ORD">-14</mrow><mrow\; data-mjx-texclass="ORD">+15</mrow>}$%, for a well-defined subset of the 8-24 keV selected sample. This is higher, albeit at a low significance level, than the type 2 fraction measured for redshift- and luminosity-matched AGNs selected by <10 keV X-ray missions.