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.