High-resolution mid-infrared spectra are presented for 155 nuclear and extranuclear regions from the Spitzer Infrared Nearby Galaxies Survey (SINGS). The fluxes for nine atomic forbidden and three molecular hydrogen mid-infrared emission lines are also provided, along with upper limits in key lines for infrared-faint targets. The SINGS sample shows a wide range in the ratio of [S III] 18.71 μm/[S III] 33.48 μm, but the average ratio of the ensemble indicates a typical interstellar electron density of 300-400 cm–3 on ~23'' × 15'' scales and 500-600 cm–3 using ~11'' × 9'' apertures, independent of whether the region probed is a star-forming nuclear, a star-forming extranuclear, or an active galactic nuclei (AGN) environment. Evidence is provided that variations in gas-phase metallicity play an important role in driving variations in radiation field hardness, as indicated by [Ne III] 15.56 μm/[Ne II] 12.81 μm, for regions powered by star formation. Conversely, the radiation hardness for galaxy nuclei powered by accretion around a massive black hole is independent of metal abundance. Furthermore, for metal-rich environments AGN are distinguishable from star-forming regions by significantly larger [Ne III] 15.56 μm/[Ne II] 12.81 μm ratios. Finally, [Fe II] 25.99 μm/[Ne II] 12.81 μm versus [Si II] 34.82 μm/[S III] 33.48 μm also provides an empirical method for discerning AGN from normal star-forming sources. However, similar to [Ne III] 15.56 μm/[Ne II] 12.81 μm, these mid-infrared line ratios lose their AGN/star-formation diagnostic powers for very low metallicity star-forming systems with hard radiation fields.