Though they do not usually own the intellectual property rights in the works that survive the initial run of the show in all its incarnations, they are essential in control and ultimately responsible for the success or failure of that run. Producers find and bring in the money, find the writer, director, venue, etc.
Anyone who has read the stage directions of a Shakespeare play can recognize how much creative input a director is allowed to interpret how the performance of those words on the page is to happen. Here's where you get to impress your friends with your fancy verbiage: The first act is the Protasis, or exposition.
Virtually all plays, as much as we rail against the way some screenwriters have turned this into a cookie-cutter, divide into what has come to be called three-act structure. Start with an Idea. Their role is to bring their particular expertise to various production problems and smooth the transition from page to stage.
Above all, you'll learn to put into practice the most fundamental axiom of playwriting: "Plays are written to be produced.