This is a baseball term used to describe the first player in the batting order overall, or for a given inning. Since they are the first up, they "lead" the order, which is why they are known as the leadoff hitter. The overall leadoff hitter is most often one of the better hitters on a team. The job of a leadoff hitter, overall or per inning, is to get on base. While hitting a solo home run is always acceptable, managers want players on base so when their strongest hitters come up, many runs can score, rather than just one.
The leadoff hitter is not usually the best hitter on the team, that player usually bats fourth overall, in what is known as the "cleanup" role. The idea being that if the first three batters, including the leadoff batter, get on base, the fourth and strongest batter will hit a grand slam and drive in all the runs. Since the term could apply to the overall lineup or a given inning, if the first player up in an inning is the ninth player in the batter order, they will be known as the leadoff hitter for the inning.