Because one of the principal themes of Hawthorne's "The Birthmark" is the conflict between nature and science, the shape of Georgiana's birthmark--a small, delicate hand--is crucial to Hawthorne's argument in this story.
At the very beginning of the story, Aylmer--a man of pure science--mentions to Georgiana that the birthmark might be removed, calling it
this slightest possible defect, which we hesitate whether to term a defect or a beauty, shocks me, as being the visible mark of earthly imperfection.
Of course, at this point, we do not know anything specific about the birthmark, but if we consider Aylmer's reaction to it, we much conclude that the birthmark might be considered a "defect" in an otherwise perfect face. More important, though, Aylmer's view that the birthmark is a defect establishes the conflict between science and nature.
This conflict is made clear when Hawthorne actually describes the nature of the birthmark and other people's reaction to it:
Its shape bore not a little similarity to the human hand, though of the smallest pygmy size. . . . Many a desperate swain would have risked life for the privilege of pressing his lips to the mysterious hand.
The shape of the birthmark becomes increasingly important as the story progresses when it is clear that Aylmer, representing science, becomes obsessed with eradicating a mark that, for other men, is an object of delight. And it's shape becomes very important as a symbol of nature.
In Hawthorne's lifetime, scientific knowledge was expanding at a great rate, but Hawthorne, drawn to transcendentalism, which emphasized the power of nature and the value of natural life, used the story to further the argument that nature was not only more powerful than man and science but also that tampering with nature could not lead to a good result. In the context of the conflict between science and nature, then, the shape of Georgiana's birthmark--the "hand" of nature--is the symbolic representation of nature's ability to create. A different shape--a cloverleaf, a circle, an unidentifiable blotch--would not have carried with it the creative power of the shape of a hand.
The birthmark's shape, then, is crucial to the conflict at the heart of the story because it represents the power of nature over mankind, a power that becomes clear when, at the end of the story, Aylmer, despite his best science, not only cannot rid Georgiana of the birthmark but also kills her in the process of its failed removal.