Third Person Omniscient
Our narrator is omniscient, all right. Here's a good example:
She had wandered, without rule or guidance, in a moral wilderness; as vast, as intricate and shadowy, as the untamed forest, amid the gloom of which they were now holding a colloquy that was to decide their fate. (18.2)
He's describing Hester's willingness to drop everything and run away with Dimmesdale. She can think these crazy thoughts because she's been outside of the community for so long. But notice that he doesn't actually tell us what she thinks. He describes it. This narrator is much more of a show-er than a tell-er, which—we'll be honest—can make the book a wee bit challenging to read. There's a lot of philosophizing, and a lot of revealing secret thoughts and actions.
Of course, we're pretty sure that Hawthorne knows what he's doing. Remember, he's got this whole story of finding the documents and the letters in the custom-house, so The Scarlet Letter is supposed to be a kind of history. But it's not just any kind of history; it's the history of two people caught up in religious and social forces way, way outside of their control. It may be about individual sin, but it's also about the epic conflict of different worldviews—so, it needs a narrator who's not afraid to get epic.