From BR Bullpen
A bat boy is a boy or girl or young man or young woman who takes care of bats and other equipment. The bat boy may fetch bats thrown by batters as they are running to first base, and may otherwise keep equipment from littering the field.
Some bat boys have gone on to become major league ballplayers.
The position is somewhat similar to the "ball boys" and "ball girls" who sometimes sit in corners of the field and fetch foul balls to remove them from the field between plays.