We present Herschel observations of 6 fine-structure lines in 25 ultraluminous infrared galaxies at z < 0.27. The lines, [O III]52 mu m, [N III] 57 mu m, [O I]63 mu m, [N II]122 mu m, [O I]145 mu m, and [C II]158 mu m, are mostly single Gaussians with widths < 600 km s(-1) and luminosities of 10(7)-10(9) L-circle dot. There are deficits in the [O I] 63/L-IR, [N II]/L-IR, [O I]145/L-IR, and [C II]/L-IR ratios compared to lower luminosity systems. The majority of the line deficits are consistent with dustier H II regions, but part of the [C II] deficit may arise from an additional mechanism, plausibly charged dust grains. This is consistent with some of the [C II] originating from photodissociation regions or the interstellar medium (ISM). We derive relations between far-IR line luminosities and both the IR luminosity and star formation rate. We find that [N II] and both [O I] lines are good tracers of the IR luminosity and star formation rate. In contrast, [C II] is a poor tracer of the IR luminosity and star formation rate, and does not improve as a tracer of either quantity if the [C II] deficit is accounted for. The continuum luminosity densities also correlate with the IR luminosity and star formation rate. We derive ranges for the gas density and ultraviolet radiation intensity of 10(1) < n < 10(2.5) and 10(2.2) < G(0) < 10(3.6), respectively. These ranges depend on optical type, the importance of star formation, and merger stage. We do not find relationships between far-IR line properties and several other parameters: active galactic nucleus (AGN) activity, merger stage, mid-IR excitation, and SMBH mass. We conclude that these far-IR lines arise from gas heated by starlight, and that they are not strongly influenced by AGN activity.