We present the discovery and extensive follow-up observations of SN 2020jfo, a Type IIP supernova (SN) in the nearby (14.5Mpc) galaxy M61. 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 are used to model the 350-day duration bolometric light curve, which exhibits a relatively short (~65 days) plateau. This implies a moderate ejecta mass (~5M_{sun}_) at the time of explosion, whereas the deduced amount of ejected radioactive nickel is ~0.025M_{sun}_. An extensive series of spectroscopy is presented, including spectropolarimetric observations. The nebular spectra are dominated by H{alpha} but also reveal emission lines from oxygen and calcium. Comparisons to synthetic nebular spectra indicate an initial progenitor mass of ~12M_{sun}_. 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 pre-discovery data are examined, but no signs of pre-cursor activity is found. Pre-explosion Hubble Space Telescope imaging reveals a probable progenitor star, detected only in the reddest band (M_F814W_~-5.8) and is fainter than expected for stars in the 10-15M_{sun}_ 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 H{alpha}-dominated spectra. This illustrates how the absence or presence of 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 H{alpha} 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 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 H{alpha}, although with an intermediate line width.