We report the results of a spectroscopic survey of reflection nebulae, aimed at studying the characteristics of the broad emission feature responsible for the extended red emission (ERE) observed in these objects. The ERE band was detected and its strength, central wavelength, and width were measured in 12 nebulae, while upper limits were recorded in 12 further objects. The maximum ERE band intensity was found to vary from object to object within an extremely narrow range while the associated scattered light intensities differed by nearly two orders of magnitude. We interpret the ERE band as arising from the photoluminescence of hydrogenated amorphous carbon (HAC) grains which become rehydrogenated and gain luminescence efficiency in narrow H2 photodissociation zones. These zones are probably thin shells around the exciting stars. Observed changes in the central wavelengths and widths of the ERE bands detected in different parts of a given nebula, and from nebula to nebula, support the HAC model and are explained as arising from variations in the degree of hydrogenation of the solid grains. The Red Rectangle, while emitting the ERE band with unusual strength, is found to be an object of a class different from ordinary reflection nebulae. The luminescent material pervades the entire nebula and appears to be produced in a local mass-loss process rather than through rehydrogenation.