Bequeathed runners scored represents the total number of runs a pitcher is charged with after he leaves the game. Or, put another way, it's the number of runners who come around to score after being left on base when that pitcher exits the game.
There is a key exception, however. If a fielder's choice eliminates one of a departed pitcher's baserunners, and the new baserunner scores, that run will still be charged to the original pitcher and not the second pitcher. This is because the second pitcher simply did his job by recording an out.
BQR-S can be a good way of assessing how much help a starting pitcher received from his bullpen. If a starting pitcher has a high percentage of his bequeathed runners score -- generally because of a shaky bullpen -- it will adversely affect his ERA, even though it's something over which he doesn't have a lot of control. (But it's important to remember that he has some control. He didn't have to allow those runners to reach base in the first place.)
Watch: Michael Feliz allows two bequeathed runners to score.
In A Call
"inherited runners who scored" (for the pitcher who is entering the game)