Aylmer's dream is a classic case of literary foreshadowing. It's also Hawthorne's way of driving home his point about the futility of separating human imperfections from our very humanity (see "What's Up with the Ending?"). Aylmer dreams that he tries to cut away his wife's birthmark. This, of course, anticipates the procedure Aylmer will attempt at the end of the narrative. In the dream he fanatically continues his attempts at removal, even at the danger of losing his wife's life. This, too, is the case at the end of the story. The heavy symbolism comes in when Aylmer sees in his dream that the birthmark goes deep, eventually settling in Georgiana's heart. Remember that the birthmark symbolizes human flaws. This is to say that Georgiana's imperfections are a very much part of her being – it would be foolish for Aylmer to imagine that he could cut them away without destroying her. As we learned in the initial description of the birthmark, it is "deeply interwoven" with Georgiana's face.
That Aylmer's dream essentially tells him what's going to happen, and that he goes forward with it anyway, raises interesting questions about Aylmer's level of self-deception in the tale. On some level, he must know that his wife is going to die; the only question is whether he recognizes this consciously and/or subconsciously when he goes forward with his experiment anyway.