Calling a play in American football is a complex process. During any one given play each of the players on a team has a different, specific, scripted role to play.
Who calls the plays? Either the head coach or the offensive coordinator. In some situations the quarterback.
How is the play communicated to the players? The quarterback may have a speaker in his helmet connecting him with the coaches. A substitute player can be sent in with the play. The play can be communicated from the sidelines via hand signals.
More and more teams are using a no-huddle offense to speed the game up. Players take their positions as quickly as possible, and get their assignments from the quarterback or from the coaches on the sideline.