We conduct a comprehensive analysis of the relationship between central galaxies and their host dark matter halos, as characterized by the stellar mass - halo mass (SM-HM) relation, with rigorous consideration of uncertainties. Our analysis focuses on results from the abundance matching technique, which assumes that every dark matter halo or subhalo above a specific mass threshold hosts one galaxy. We provide a robust estimate of the SM-HM relation for 0 < z < 1 and discuss the quantitative effects of uncertainties in observed galaxy stellar mass functions (GSMFs) (including stellar mass estimates and counting uncertainties), halo mass functions (including cosmology and uncertainties from substructure), and the abundance matching technique used to link galaxies to halos (including scatter in this connection). Our analysis results in a robust estimate of the SM-HM relation and its evolution from z=0 to z=4. The shape and evolution are well constrained for z < 1. The largest uncertainties at these redshifts are due to stellar mass estimates (0.25 dex uncertainty in normalization); however, failure to account for scatter in stellar masses at fixed halo mass can lead to errors of similar magnitude in the SM-HM relation for central galaxies in massive halos. We also investigate the SM-HM relation to z = 4, although the shape of the relation at higher redshifts remains fairly unconstrained when uncertainties are taken into account. We find that the integrated star formation at a given halo mass peaks at 10-20% of available baryons for all redshifts from 0 to 4. This peak occurs at a halo mass of 71011M_ at z = 0 and this mass increases by a factor of 5 to z = 4. At lower and higher masses, star formation is substantially less efficient, with stellar mass scaling as M* is asymptotically equivalent to M2.3 h at low masses and M* is asymptotically equivalent to M0.29 h at high masses. The typical stellar mass for halos with mass less than 1012 M_ has increased by 0.3 0.45 dex for halos since z is approximately equal to 1. These results will provide a powerful tool to inform galaxy evolution models. Show Partial Abstract