This is a phrase used in baseball to refer to a pitcher who comes into the game when the game has essentially been decided. The team putting the mop-up pitcher most likely has a large lead and is not in any danger of losing the game. It can be a risky move if the lead is not large enough. It would only be used during the regular season or in pre-season, never in the playoffs or World Series. This pitcher is almost always a rookie or other inexperienced player. They are put into the game to gain experience and pitch in a game that matters, without putting the game on the line.
The phrase, "mop-up," refers to the cleaning up the pitcher is doing by finishing the game. These pitchers usually jump at the chance to pitch in a real game, despite its unimportance to the actual game. It also helps the team because it allows the regular starter or reliever to rest. Managers will often take this into consideration late in the season when playoff positioning is already set.