We report on the first 10 identifications of sources serendipitously detected by the Nuclear Spectroscopic Telescope Array ( NuSTAR ) to provide the first sensitive census of the cosmic X-ray background source population at ≳ 10 keV. We find that these NuSTAR -detected sources are ≈ 100 times fainter than those previously detected at ≳ 10 keV and have a broad range in redshift and luminosity ( z = 0.020-2.923 and L10-40 keV ≈ 4 × 1041 -5 × 1045 erg s-1); the median redshift and luminosity are z ≈ 0.7 and L 10-40 keV ≈ 3 × 1044 erg s-1, respectively. We characterize these sources on the basis of broad-band ≈ 0.5-32 keV spectroscopy, optical spectroscopy, and broad-band ultraviolet-to-mid-infrared spectral energy distribution analyses. We find that the dominant source population is quasars with L10-40 keV > 1044 erg s-1, of which ≈ 50% are obscured with NH ≳ 1022 cm-2. However, none of the 10 NuSTAR sources are Compton thick (NH ≳ 1024 cm-2) and we place a 90% confidence upper limit on the fraction of Compton-thick quasars (L10-40 keV > 1044 erg s-1) selected at ≳ 10 keV of ≲ 33% over the redshift range z = 0.5-1.1. We jointly fitted the rest-frame ≈ 10-40 keV data for all of the non-beamed sources with L10-40 keV > 1043 erg s-1 to constrain the average strength of reflection; we find R <1.4 for Γ = 1.8, broadly consistent with that found for local active galactic nuclei (AGNs) observed at ≳ 10 keV. We also constrain the host-galaxy masses and find a median stellar mass of ≈ 1011 M⊙, a factor ≈ 5 times higher than the median stellar mass of nearby high-energy selected AGNs, which may be at least partially driven by the order of magnitude higher X-ray luminosities of the NuSTAR sources. Within the low source-statistic limitations of our study, our results suggest that the overall properties of the NuSTAR sources are broadly similar to those of nearby high-energy selected AGNs but scaled up in luminosity and mass.