It depends on the team ... Coaches, offensive coordinators, and/or the quarterback call plays. On the defensive side there are formations called, the equivalent of plays for the defense. On the offense, if a play is called but the defensive formation is not conducive for that play, the QB can call an audible at the line.
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The coaches usually call the plays. If not, it's the quarterback for the offense and normally the middle line backer on defense.
The person who creates the play. In some cases this can be the coach, quarterback, or who ever comes up with a decent play such as the team's wide receiver, snapback, or defensive line.