We discuss the spectral analysis of a sample of 63 active galactic nuclei (AGN) detected above a limiting flux of S 8 24 keV 7 10 14 =´-( -) erg s cm 1 2 --in the multi-tiered NuSTAR extragalactic survey program. The sources span a redshift range z 0 2.1 = -(median z 0.58 á ñ = ). The spectral analysis is performed over the broad 0.5-24keV energy range, combining NuSTAR with Chandra and/or XMM-Newton data and employing empirical and physically motivated models. This constitutes the largest sample of AGN selected at 10 keV > to be homogeneously spectrally analyzed at these flux levels. We study the distribution of spectral parameters such as photon index, column density (N H ), reflection parameter (R), and 10-40keV luminosity (L X ). Heavily obscured ( N log cm 23AGN constitute ∼25% (15-17 sources) and ∼2-3% (1-2 sources) of the sample, respectively. The observed N H distribution agrees fairly well with predictions of cosmic X-ray background population-synthesis models (CXBPSM). We estimate the intrinsic fraction of AGN as a function of N H , accounting for the bias against obscured AGN in a flux-selected sample. The fraction of CT AGN relative to N log cm 20 24AGN is poorly constrained, formally in the range 2-56% (90% upper limit of 66%). We derived a fraction ( f abs ) of obscured AGN ( N log cm 22 24as a function of L X in agreement with CXBPSM and previous z 1 < X-ray determinations. Furthermore, f abs at z 0.1 0.5 = -and L log erg s 43.6 44.3agrees with observational measurements/trends obtained over larger redshift intervals. We report a significant anti-correlation of R with L X (confirmed by our companion paper on stacked spectra) with considerable scatter around the median R values.