A catcher is given a passed ball if he cannot hold onto a pitch that -- in the official scorer's judgment -- he should have, and as a result at least one runner moves up on the bases. Passed balls have commonality with wild pitches, as both allow a runner to advance on his own without a stolen base. However, there is a key difference: A passed ball is deemed to be the catcher's fault, while a wild pitch is deemed to be the fault of the pitcher.
A passed ball is not recorded as an error, but when a run scores as the result of a passed ball, it does not count as an earned run against a pitcher. (In cases where this is in question, the official scorer must reconstruct the inning, and if the run would not have scored without the passed ball, that run is deemed unearned.) If a runner advances on a passed ball, he is not credited with a stolen base.
After a strikeout, if the catcher fails to catch the third strike, and the batter reaches first base safely as a result, either a passed ball or a wild pitch must be awarded. In the instance of a wild pitch, that baserunner could count against a pitcher's ERA, but in the instance of a passed ball, he cannot.