A play sheet in football is a list of plays used by the coach to call plays from. Each play sheet is incredibly detailed and is made up of plays that are selected with the other team's weaknesses in mind.
Whichever coach is in charge of play calling (in most cases the offensive coordinator) holds the play sheet during the game. They read it and find plays that best fit the situation the offense is in. It has plays grouped together based on the yardage needed and current down.
While the defensive coordinator doesn't have a sheet for offensive plays, they do carry around a sheet with defensive schemes on it. Those schemes and coverages are also grouped into categories such as blitzes, zone coverages, and man to man coverage.
Play sheets help a football team largely with organization. With only 40 seconds between each play a coaching staff must decide quickly what is the best play to call. By organizing each of their possible plays into easy to read categories, the coach can call up a play that fits the situation very quickly.