In MLB there is a lot of pre-game study as to what pitches to throw to what batters during a game based on their hitting patterns.
Who is most responsible for knowing these and who calls the pitches during a game?
Pitcher? Catcher? Pitching coach? Head coach?
In the past the manager did not call pitches but now it happens more often. Most of the time the catcher calls it but of course the pitcher can change it by shaking his head.
Some interesting links:
I’m used to catchers calling the game with lots of prep from the advanced team and studying stats, but I guess there’s been a movement in recent years for the manager or pitching coach to call many pitches. The catcher I’ve seen the most is Jason Varitek of the Red Sox, and he appears to call most of the pitches in his games. It may be quite different for younger catchers or ones that aren’t as familiar with his pitchers.
The catcher calls nearly all the pitches, though things like intentional walks are called by the manager.
It’s difficult to set up a system where the manager can relay information to the catcher without the other team figuring it out. Intentional walks don’t need to be kept secret – the batter (usually) can’t do anything even if he knows what’s coming – so the manager could just yell, “walk him.”
Basically, the catcher has to remember what types of pitches will work with each batter. They also consult with the manager or pitching coach between innings about pitch selection.
Looking at the other team’s signs is prevented mostly by it being Just Not Done. A batter, for instance, who looks back at the catcher can expect the next pitch to come in high and very tight.
If you ever see the catcher’s signal is the middle finger only… either the umpire is ‘accidentally’ going to get hit or the batter has one coming into his ear hole. 
“On this next pitch, I want you to hit the mascot.”
“And when you speak of me, speak well.” 
“Buy low, sell high.”
“Throw some ground balls, it’s more democratic.”