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We report the results of a program of long-slit spectroscopy, narrow-band imaging, and VLA mapping of the optical emission-line gas and nonthermal radio continuum emission associated with the dominant galaxy in nine galaxy clusters with cooling flows. These data provide important information about the physical, dynamical, and chemical state of the cooling flow material. All the emission-line nebulae we observed are very strongly centrally concentrated with typical characteristic radii of 5 kpc or less. If (as recent analyses of the X-ray data have suggested) the mass inflow rate (M^dot^) varies linearly with radius in cooling flows out to large radii, then the Hα surface brightness does not trace the cooling rate. Our principal kinematic result is that the velocity field of the emission-line gas provides little evidence for organized rotation or shear. This result can be understood if the emission-line filaments condense out of an intracluster medium (ICM) in which angular momentum has been transported radially outward. Lines are broad throughout the nebulae, but line widths systematically decrease with increasing radius, consistent with infall of the emission-line clouds at about the free-fall velocity in the galaxy potential (as expected if the clouds form in the supersonic innermost region of the cooling flow). If the intercloud medium is also infalling, its infall rate is rather close to the values for the cooling/inflow rate inferred from the X-ray data (M^dot^). Alternatively, the ICM may be highly turbulent, especially at small radii. We have also measured electron densities over spatially resolved regions in seven cases. The central densities (r < 1 kpc) are typically a few hundred per cubic centimeter, implying central pressures of about (1-2) x 10^-9^ dynes cm^-2^. These pressures are about a factor of 10 larger than those estimated for the X-ray gas on somewhat larger spatial scales. At radii of several kiloparsecs the densities and pressures have usually dropped by a factor of a few compared with the central values. Spatial variations in the emission-line ratios in the nebulae agree better with shock models than with photoionization models. The cooling flow nebulae fall into two distinct spectral classes, with membership being strongly related to X-ray luminosity, emission-line luminosity, and mass accretion rate. The spectral differences between the two classes cannot be readily attributed solely to variations in shock speed or ionization parameter, and so we suggest that variations in chemical abundances may be responsible. Our spectrum of the 3C 295 nebula reveals high-excitation gas, quite unlike the other cases. This, and the very high ratio of Hα luminosity to mass accretion rate, lead us to suggest that the gas in 3C 295 is photoionized by an active galactic nucleus (AGN). We have failed to detect the [Fe x] λ6374 emission line in any of the nebulae. For solar values of the Fe:Ne:O abundance ratios, the upper limits on the cooling accretion rate are 10-1000 M_sun_ yr^-1^ (comparable in some cases to the rates inferred from the X-ray data). We discuss the various possible mechanisms for heating and ionizing the emission-line nebulae. The "standard" picture in which repressurizing shocks are driven into cooling clouds by the hot ICM is viable only if each cooling H atom typically emits about 10-100 Hα photons and if most of the mass in the flow drops out within the region of the nebula (1-10 kpc scale). Heating by X-ray from the ICM appears to be energetically adequate, provided that the clouds are optically thick to soft X-rays and cover a significant fraction of the central X-ray nebula. This is reasonable if the emission-line clouds contain significant amounts of neutral gas (for which there is clear observational evidence). We can firmly rule out photoionization by an AGN on several grounds. If massive stars form in the cooling flow, as recent data have suggested, photoionization of the nebulae by OB stars may be energetically feasible. However, the emission-line ratios in the spectra of the nebulae are inconsistent with this hypothesis. For a normal initial mass function, shock-heating by associated supernovae can provide at most only 10% as much energy input as the Lyman continuum from OB stars, and would not be expected to alter the emission-line spectrum strongly. The estimated kinetic energy flux in the ICM is approximately equal to the emission-line luminosity in nebulae spanning 3 orders of magnitude in luminosity. The emission-line clouds may then be heated by the dissipation of the kinetic energy of the ICM, but only if this energy is converted into emission-line luminosity with an uncomfortably large efficiency (~ 100%). We also consider the possibility that the interaction between the radio source and the ICM may be important in heating the optical emission-line filaments. In particular, we emphasize that Coulomb heating of the emission-line clouds by the population of relativistic electrons responsible for the radio emission, and heating by magnetic field reconnection, may both be energetically important. In support of these ideas, we summarize the relationship between the emission-line gas, the radio source, and the ICM in fooling flow clusters. We find that the radio power is correlated with the emission-line luminosity (with the latter being typically 10 times larger). The radio and emission-line plasma often appear to be approximately cospatial, and the optical emission lines are significantly broader in regions of bright radio emission. The minimum pressures in the radio plasma area similar to those inferred for the X-ray gas. The associated radio sources are generally small and only weakly polarized, expected consequences of the presence of a dense, high-pressure ambient medium (the ICM). Finally, the radio power and the accretion rate M^dot^_x_ are correlated (but only weakly).