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We present HST/ACS narrowband images of a low-z sample of 19 3C radio galaxies to study the Hα and [O iii] emissions from the narrow-line region. Based on nuclear emission-line ratios, we divide the sample into high- and low-excitation galaxies (HEGs and LEGs). We observe different line morphologies, extended [O iii] emission, large [O iii]/Hα scatter across the galaxies, and a radio-line alignment. In general, HEGs show more prominent emission-line properties than LEGs: larger, more disturbed, more luminous, and more massive regions of ionized gas with slightly larger covering factors. We find evidence of correlations between line luminosities and (radio and X-ray) nuclear luminosities. All of these results point to a main common origin, the active nucleus, which ionizes the surrounding gas. However, the contribution of additional photoionization mechanisms (jet shocks and star formation) is needed to account for the different line properties of the two classes. A relationship between the accretion, photoionization, and feedback modes emerges from this study. For LEGs (hot-gas accretors), the synchrotron emission from the jet represents the main source of ionizing photons. The lack of cold gas and star formation in their hosts accounts for the moderate ionized-gas masses and sizes. For HEGs (cold-gas accretors), an ionizing continuum from a standard disk and shocks from the powerful jets are the main sources of photoionization, with a contribution from star formation. These components, combined with the large reservoir of cold/dust gas brought from a recent merger, account for the properties of their extended emission-line regions. [ABSTRACT FROM AUTHOR]