We present the discovery and extensive follow-up observations of SN 2020jfo, a Type IIP supernova (SN) in the nearby (14.5 Mpc) galaxy M 61. Optical light curves (LCs) and spectra from the Zwicky Transient Facility (ZTF), complemented with data from Swift/UVOT and near-infrared photometry is presented. These were used to model the 350-day duration bolometric light curve, which exhibits a relatively short (∼65 days) plateau. This implies a moderate ejecta mass (∼5 M⊙) at the time of explosion, whereas the deduced amount of ejected radioactive nickel is ∼0.025 M⊙. An extensive series of spectroscopy is presented, including spectropolarimetric observations. The nebular spectra are dominated by Hα, but also reveal emission lines from oxygen and calcium. Comparisons to synthetic nebular spectra indicate an initial progenitor mass of ∼12 M⊙. We also note the presence of stable nickel in the nebular spectrum, and SN 2020jfo joins a small group of SNe that have inferred super-solar Ni/Fe ratios. Several years of prediscovery data were examined, but no signs of precursor activity were found. Pre-explosion Hubble Space Telescope imaging reveals a probable progenitor star, detected only in the reddest band (MF814W ≈ −5.8) and it is fainter than expected for stars in the 10−15 M⊙ range. There is thus some tension between the LC analysis, the nebular spectral modeling, and the pre-explosion imaging. To compare and contrast, we present two additional core-collapse SNe monitored by the ZTF, which also have nebular Hα-dominated spectra. This illustrates how the absence or presence of an interaction with circumstellar material (CSM) affect both the LCs and in particular the nebular spectra. Type II SN 2020amv has a LC powered by CSM interaction, in particular after ∼40 days when the LC is bumpy and slowly evolving. The late-time spectra show strong Hα emission with a structure suggesting emission from a thin, dense shell. The evolution of the complex three-horn line profile is reminiscent of that observed for SN 1998S. Finally, SN 2020jfv has a poorly constrained early-time LC, but it is of interest because of the transition from a hydrogen-poor Type IIb to a Type IIn, where the nebular spectrum after the light-curve rebrightening is dominated by Hα, although with an intermediate line width.