We have obtained moderate-resolution (R similar to 3000) spectra of the Orion bar and Orion S regions at J (1.25 mu m), H (1.64 mu m), and K (2.2 mu m). Toward the bar, the observations reveal a large number of H-2 emission lines that, when compared to model predictions of Draine & Bertoldi, are indicative of a high-density photodissociation region (PDR) (n(H) = 10(6) cm(-3), chi = 10(5), T-0 = 1000 K) rather than of shocked material. Behind the bar and into the molecular cloud, the H-2 spectrum again matches well with that predicted for a dense PDR (n(H) = 10(6) cm(-3)) but with a lower temperature (T-0 = 500 K) and UV field strength (chi = 10(4)). The H-2 spectrum and stratification of near-IR emission lines (O I, H I, [Fe II], [Fe III], H-2) near Orion S imply the presence of a dense PDR with an inclined geometry in this region (n(H) = 10(6) cm(-3), chi = 10(5), T-0 = 1500 K). The extinction measurements toward the bar (A(K) similar to 2.6) and Orion S (A(K) similar to 2.1) H-2 emission regions are much larger than expected from either face-on (A(K) similar to 0.1) or edge-on (A(K) similar to 1) homogeneous PDRs, indicating that clumps may significantly affect the structure of the PDRs. In addition, we have observed the strongest similar to 30 near-IR He I emission lines, many of which have not been detected previously. There is good agreement between most observed and theoretical He I line ratios, while a few transitions with upper levels of n P-3 (particularly 4(3)P-3(3)S 1.2531 mu m) are enhanced over strengths expected from collisional excitation. This effect is possibly due to opacity in the UV series n P-3-2(3)S. We also detect several near-IR [Fe II] and [Fe III] transitions with line ratios indicative of low densities (n(e) similar to 10(3)-10(4) cm(-3)), whereas recent observations of optical [Fe II] emission imply the presence of high-density gas (n(e) similar to 10(6) cm(-3)). These results are consistent with a model in which high-density, partially-ionized gas is the source of the iron transitions observed in the optical, while low-density, fully-ionized material is responsible for the near-IR emission lines.