Unorthodoxy: your priest is more likely to resist being turned into a heretic, when trying a heretic, or killed, when trying a witch, if he has high orthodoxy. If he is unorthodox, then he has a higher chance of being either converted or killed. Higher unorthodoxy also means he is more likely to turn into a heretic on his own.
Eligibility: obviously he is more likely to be chosen as a preferati if he is more eligible, even if another priest has the same or slightly higher piety rating.
Purity: your priest is less likely to be corrupted, which incidentally is the anti-trait. I don't know the exact effect of this, as it seems to be another trait that has a hidden effect; but it seems that my priests survive heretic/witch trial failures more with this trait. I think it also helps to get your priest chosen as a cardinal over another priest with the same or slightly higher piety rating. I would like to think that a corrupt Pope is also easier to manipulate (throwing money at him), but that doesn't seem to be the case at all.