As I understand it, the crew chief is usually the most senior member of the four man umpire crew. He acts as liaison between the crew and the league. He is the one who reports to league officials about any issues during a game, such as delays, protests, ejections, etc.
However, he has no more authority during a game than any other umpire. That rests with the umpire in chief or UIC. Who is usually the plate umpire. The UIC generally rotates between the four umpires. That is, the first game you are behind the plate. The next, you are at first base. Then second, third and back to the plate.
The UIC does have additional authority. He/she decides when to start the game and, if necessary, when to suspend the game. However the principle authority he or she has is that when two umpires make differing calls on the same play, the UIC decides which was in the better position to see the play and makes the decision.
So the crew chief only has more authority than the other umpires when it is his/her turn in the rotation to be the plate umpire or umpire in chief.
SOURCE: Quora