Who is the narrator, can she or he read minds, and, more importantly, can we trust her or him?
Third-Person Omniscient
Technically, since this is a play, there isn't really a narrator in the same way there's a narrator for a novel, short story, or poem. The narration is really in the descriptions at the start of each part of the trilogy that describe characters and parts of the setting, like when we read that Josiah Borden's wife is "a typical New England woman of pure English ancestry, with a horse face, buck teeth, and big feet." (Shmoop's New England-based employees are definitely gonna write in about that.) Narration is also in the stage directions that are meant to give actors hints about how to speak or react to certain lines.
Still, we get first-hand access to information that plenty of other characters don't, and O'Neill has us moving around across time and space like we're all Marty McFly in Back to the Future. That's a sure bet we're dealing with your classic third-person omniscient narrator.