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We present the first comprehensive release of photometric redshifts (photo-z's) from the Cosmic Assembly Near-Infrared Deep Extragalactic Legacy Survey (CANDELS) team. We use statistics based upon the Quantile-Quantile (Q--Q) plot to identify biases and signatures of underestimated or overestimated errors in photo-z probability density functions (PDFs) produced by six groups in the collaboration; correcting for these effects makes the resulting PDFs better match the statistical definition of a PDF. After correcting each group's PDF, we explore three methods of combining the different groups' PDFs for a given object into a consensus curve. Two of these methods are based on identifying the minimum f-divergence curve, i.e., the PDF that is closest in aggregate to the other PDFs in a set (analogous to the median of an array of numbers). We demonstrate that these techniques yield improved results using sets of spectroscopic redshifts independent of those used to optimize PDF modifications. The best photo-z PDFs and point estimates are achieved with the minimum f-divergence using the best 4 PDFs for each object (mFDa4) and the Hierarchical Bayesian (HB4) methods, respectively. The HB4 photo-z point estimates produced $σ_{\rm NMAD} = 0.0227/0.0189$ and $|Δz/(1+z)| > 0.15$ outlier fraction = 0.067/0.019 for spectroscopic and 3D-HST redshifts, respectively. Finally, we describe the structure and provide guidance for the use of the CANDELS photo-z catalogs, which are available at https://archive.stsci.edu/hlsp/candels.