We present an analysis of the z ~ 0 morphology-environment relation for 911 bright (M_B < -19) galaxies, matching classical RC3 morphologies to the SDSS-based group catalog of Yang et al. We study how the relative fractions of spirals, lenticulars, and ellipticals depend on halo mass over a range of 10^11.7-10^14.8 h^-1 Msol. We pay particular attention to how morphology relates to central (most massive) vs satellite galaxy status. The fraction of galaxies which are elliptical is a strong function of stellar mass; it is also a strong function of halo mass, but only for central galaxies. We interpret this in a scenario where elliptical galaxies are formed, probably via mergers, as central galaxies within their halos; satellite ellipticals are previously central galaxies accreted onto larger halos. The overall fraction of S0 galaxies increases strongly with halo mass, from ~10% to ~70%. We find striking differences between the central and satellites: 20+/-2% of central M_* > 10^10.5 Msol galaxies are S0 regardless of halo mass, but satellite S0 galaxies are only found in massive (> 10^13 h^-1 Msol) halos, where they are 69+/-4% of the M_* > 10^10.5 Msol satellite population. This suggests two channels for S0 formation: one for central galaxies, and another which transforms lower mass (M_* <~ 10^11 Msol) accreted spirals into satellite S0 galaxies in massive halos. Analysis of finer morphological structure (bars and rings in disk galaxies) shows some trends with stellar mass, but none with halo mass; this is consistent with other recent studies which indicate that bars are not strongly influenced by galaxy environment. Radio sources in high-mass central galaxies are common, similarly so for elliptical and S0 galaxies, with a frequency that increases with halo mass. Emission-line AGN (mostly LINERs) are more common in S0s, but show no strong environmental trends (abridged).
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