Nathaniel Hawthorne was a part of the American Romantic movement (1830-1860), more specifically the Dark Romantic (or Gothic). Romantics, and their texts, tended to idealize the individual, the imagination, the emotional, and the exotic. Hawthorne's "The Birthmark" is representative of the period based upon the imagery and themes of the text.
Georgiana's birthmark represents two very different things: mortality/imperfection and original sin. Both of these ideas speak to the thoughts of the Romantic. Imperfection makes a human mortal. Being mortal means that one will die, and death is a typical subject within the Romantic genre. This speaks to the idea that all will die; death is simply a part of nature (another important idea within Romantic literature). (On a side note, the Romantics believed nature to possess power which was superior to that of man.)
In regards to the birthmark representing original sin, all humans are born with this (original sin). The birthmark symbolizes the sin of vanity. As soon as the birthmark fades, Georgiana dies. The vanity of both her and Alymer (her husband) illustrate mankind's vanity and need for perfection. Given that mankind is not perfect, given they possess original sin, the removal of sin (the birthmark) removes one's life as well (since it is only through death that sin can be removed).