I would say the initial conflict that shapes "The Birthmark" is person versus person: Aylmer's disapproval of his wife's birthmark, which surprises and upsets Georgiana. What's important here, however, is that this initial conflict between two people, first expressed when Aylmer asks Georgiana about removing the mark, will eventually be transformed and internalized so that Georgiana herself comes to despise the mark (eventually even more than her husband had initially despised it). Thus, while these two factors can be expressed as two separate conflicts, it is important to note the close connection between them, with the one informing and creating the other. The external disapproval of her husband has resulted in an internal crisis within Georgiana herself, which will shape the rest of the plot.
The third critical conflict, also internal (and one which, in many respects, hangs over the entire story as a whole), revolves around Aylmer and his own obsession with perfection. This is ultimately what creates his initial dissatisfaction with the birthmark, and it will later be reflected in his various experimentations, by which he seeks to cure Georgiana of that imperfection. His is a very dangerous obsession (both destructive and self-destructive), which results in the death of his wife.
In the story's opening, the first conflict is introduced with these words: "He had devoted himself, however, too unreservedly to scientific studies ever to be weaned from them by any second passion." The narrator establishes that Aylmer's first and perhaps truest love is for science. When he marries, the reader understands that his love for science and love for his wife, Georgiana, will be a source of conflict.
The second conflict arises when Aylmer very undiplomatically asks Georgiana if she has ever thought of having her birthmark removed. She is offended, flushes with anger, and then asks him why he ever married her. Thus begins the story's second conflict. Their marriage becomes strained, as he cannot hide his revulsion and she becomes insecure as a result.
Eventually, Georgiana decides that she wants Aylmer to remove the birthmark, because she sees him shudder when he looks closely at it. This is another conflict in the story; Georgiana now cares more about her husband's opinion of her "imperfection" than her own comfort with it. This begins the section of the story in which Aylmer grapples with being in over his head insofar as the medical knowledge he would need to safely remove the birthmark, and Georgiana puts her life in his hands with tragic results.
It's been a while since I've read this story, so let me start you with these two conflicts:
The main conflict in this story is indicated by the title--Georgina's birthmark. She is beautiful, charming, in touch with nature, and completely well-adjusted. She has always seen her birthmark as a blessing, and in fact, she looks at it (a small hand-shape) as the place where she was touched by an angel. Her husband, however, can not find any beauty in it. Rather, he sees it as a flaw in his wife's appearance. Quite sad, really, that he can't seem to accept her for the way she is.
A second conflict is that he is a scientist of questionable reputation. Every experiment he has attempted has been a collossal failure. Yet, in spite of this, he has taken it upon himself to trust his ability as a scientist to successfully remove Georgina's birthmark without damage to his lovely wife. Of course, as foreshadowed by his string of failures, he ends up killing her. Ironically, he is successful in removing the birthmark, but as it is her essence, or rather the "heart" of her, removing it also removes her ability to live.