This is an expression which is mostly used in informal contexts.
A person who calls the shots is the boss. He/she decides what must be done. The individual is in total control of the situation.
There are two explanations regarding the origin of this idiom.
The first one is related to the world of target shooting. A good marksman, before firing his gun, usually informs the audience which target he will be hitting.
The other explanation is from billiards. In this game, the player announces in advance which ball he is going to hit next and which pocket it will be going into.
1. Why do you blame me? I wasn't the person calling the shots.
2. The opposition alleges that it's not the prime minister but the president of his political party, who is calling the shots.
3. The captain of the ship has the right to call the shots. He will decide the course of the journey.