From BR Bullpen
An inherited runner is a baserunner who is already on base when a relief pitcher enters the game in the middle of an inning. If this runner then scores a run, it will be credited to the record of the outgoing pitcher, and not of the relief pitcher.
However, a relief pitcher's objective is to prevent inherited runners from scoring, and a failure to do this effectively is a blemish on his record, even though it is not captured in statistics such as his win-loss record or ERA. Therefore, more advanced statistics on relief pitching will take note of how many inherited runners a relief pitcher has had to deal with, and how many were able to score.
Even this is not fully satisfactory, as not all inherited runners are the same: a relief pitcher who enters the game with none out and a runner on third base has a much more difficult task ahead of him than one who comes in with a runner on first and two outs. Therefore, some sabermetricians have developed measures of a relief pitcher's effectiveness based on the run expectation associated with the situation when he enters the game (i.e. the number of runs that would, on average, be scored in the remainder of an inning starting from this point). Thus, the relief pitcher who allows the inherited runner to score from third base over the course of a full inning is hardly penalized, whereas the one who lets the runner score from first with two outs is considered to have failed significantly.