The narrator of "The Birthmark" is allowed access to both Aylmer and Georgiana's thoughts. When learn, for example, that Aylmer perceives of the birthmark as "the symbol of his wife's liability to sin, sorrow, decay, and death," and also get the details of Georgiana's emotional reaction to her husband's opinion (she dreaded lest a gush of tears should interrupt what she had to say") (8, 13). There is, however, the odd moment or two where the narrator throws his omniscience out:
We know not whether Aylmer possessed this degree of faith in man's ultimate control over Nature. (1)
This is such an interesting line, because the narrator confesses that he is in fact ignorant of all the facts. It's odd, because for most of the narrative he maintains omniscience, with access to both Aylmer and Georgiana's thoughts. It's possible that Hawthorne is making a point about the limitations of human knowledge. For more on this, you'll want to check out "In a Nutshell."
The other big thing to talk about when it comes to this narrator is that he sure seems to have all the big moralistic opinions. That is, it wouldn't be unreasonable to think of the narrator as being Hawthorne's own voice. And he certainly doesn't hold back from the moralizing. For examples and more, see our discussion of "Tone."