Earned vs. Unearned Run
ERAn earned run is charged to the pitcher; an unearned run scored only because of an error or passed ball is not.
An earned run is the pitcher's responsibility and counts toward their ERA. An unearned run scored only because of a defensive mistake, an error or a passed ball, is the defense's fault and does not count against the pitcher.
Determining which is which requires reconstructing the inning as if the errors and passed balls had not happened, then counting how many runs would have scored anyway. Those are earned; the rest are unearned. If the defense should already have had three outs but an error extended the inning, every run after that point is unearned, even on clean hits.
This is one of the most complex calculations in scoring, especially with relief pitchers inheriting runners, which is exactly the kind of bookkeeping a scoring engine should handle for you.
Bleacher Notes reconstructs the inning to separate earned from unearned runs, so a pitcher's ERA isn't penalized for the defense's mistakes.
What makes a run unearned?
A run is unearned when it scored only because of an error or passed ball: if you replay the inning without the misplay and the run wouldn't have scored, it's unearned and doesn't count against the pitcher's ERA.
Let the scorebook keep itself.
Bleacher Notes scores every play from a single spoken sentence, applying these rules for you. In beta now, free for parents.