When exactly does a pitcher *earn* an earned run?

And conversely, when is he let off the hook with an *unearned* run? What if, for instance, a batter makes a bunt hit single -- but reaches second on a fielding error (I once saw a third baseman field the ball cleanly, only to make a wild throw to first)? If he comes around to eventually score, would it count as an *unearned* run? The thing is: one could make the case that the batter reached at least FIRST base on a genuine hit given up by the pitcher. It's just that he advanced to second on the wild throw ...