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We present optical spectroscopic data on the ionized nebulae associated with 14 galaxies that are strong far-infrared emitters ("far-IR galaxies," or "FIRGs"). This sample includes the three nearest FIRGs (M82, NGC 253, and NGC 4945), three of the nearest and most famous examples of very luminous FIRGS with bolometric luminosities of nearly 10^12^ L_sun_ (NGC 3690/IC 694, Apr 220, and NGC 6240), and the most luminous known FIRG (IRAS 00182-7112, L_IR_ ~ 10^13^ L_sun_). We find that our data provide both qualitative and quantitative support for the "superwind" model in which the kinetic energy provided by supernovae and winds from massive stars in a central starburst drives a large-scale outflow that can shock heat and accelerate ambient interstellar or circumgalactic gas. For the three nearest FIRGs (all disk galaxies seen nearly edge-on) we find that clear kinematic signature of an outflow along the galaxy's minor axis. Kiloparsec-scale regions well out of the disk are characterized by double emission-line profiles with line splitting of 200-600 km s^-1^. The emission-line gas evidently lies along the surface of an expanding bubble or outflowing conelike structure. Similar structures are present in the edge-on FIRG NGC 3079 and in the brighter "Hα bubble" in Arp 220. The gas in both NGC 6240 and IRAS 00182-7112 is very violently disturbed kinematically (lines having FWHW 600-1000 km s^-1^ over galaxy-scale regions). The sizes, velocities, and kinematic/morphological structure of these outflows agree well with the superwind model, which predicts that the winds fluid will inflate an expanding cavity in the ISM of the FIRG, with optical line emission coming from the thin outer shell of shock- heated, swept-up ambient gas. We also find that the FIRG nebulae are highly overpressured (by three to four orders of magnitude) relative to the ISM of the Milky Way, with the pressures dropping systematically with distance from the nucleus. The radius at which a fiducial pressure occurs is proportional to the square root of the IR luminosity. In the best- studied cases, the shapes of the radial density/pressure profiles are shallow at radii of less than few hundred parsecs and steep at radii greater than 1 kpc. These results are all in excellent agreement with theoretical predictions of the superwind model (relatively constant, static thermal pressure inside the region of energy/mass injection and ram pressure proportional to the star-formation rate at large radii). The shapes of the pressure profiles are inconsistent with a wind driven by an AGN. The sizes of the mass/energy injection regions derived from the pressure profiles agree well with directly measured starburst radii, and the measured pressures at large radii are close to the predicted ram pressures for starburst-driven superwinds. We find that superwinds are energetically adequate to power both the observed optical and X-ray nebulae, and also show that the relative emission-line intensities and their radical variations are consistent with ionization by wind-driven shocks, but not with photoionization by normal O stars or an AGN. We discuss the possible astrophysical implications of superwinds. At the present epoch, a typical superwind may inject 10^8^ M_sun_ of metals and 10^58^ ergs into the intergalactic medium (IGM) over its estimated lifetime of 10^7^ yr. The local luminosity function for FIRGs implies that about 10^9^ M_sun_ of metals and 10^59^ ergs could be injected, on average, per L_8_ galaxy over a Hubble time, even with no cosmic evolution in the superwind rate. Superwinds may play important or even dominant roles in the metal enrichment and heating of both the intracluster medium and general intergalactic medium. Superwinds may make an important contribution to the cosmic X-ray background, and their relationship to the QSO absorption-line phenomenon should be explored. We also speculate that superwinds may represent a phase in the evolution of a FIRG to a QSO/AGN and of a disk- disk galaxy merger to an elliptical galaxy. Finally, we emphasize that powerful FIRGs may be reasonable approximations to galaxies in formation. If so, then superwinds may be important during the process of galaxy formation, with particular relevance to the "explosions" or the "feedback" mechanisms suggested by Ostriker, Cowie, Ikeuchi, Dekel and Silk, White and others.