Most good stories start with a fundamental list of ingredients: the initial situation, conflict, complication, climax, suspense, denouement, and conclusion. Great writers sometimes shake up the recipe and add some spice.
Initial Situation
Aylmer is thoroughly devoted to science, and also recently married.
Before the birthmark even comes into play, we get the classic set-up: a little background information on our characters, and a subtle introduction of the story's themes (like science, Nature, discovery, knowledge, ambition).
Conflict
Aylmer isn't into Georgiana's birthmark and wants to remove it.
Given the title, we can be fairly sure when this topic arises that it is the focus, and indeed central conflict, of our story. It's well-suited to the job since, like most conflicts, it brings with it discontent and possible strife on the part of the characters.
Complication
Taking off Georgiana's birthmark isn't as simple as it seems.
Hawthorne warns us that the birthmark is deeply interwoven into Georgiana's countenance, and of course Aylmer's own dream anticipates that the birthmark goes deeper than the surface. There's also quite a bit of talk about Nature guarding her secrets, and of Aylmer's many failures as a scientist.
Climax
Georgiana drinks the liquid (whatever it is).
OK, so Hawthorne is a little fuzzy about what exactly Aylmer cooked up, but the dead give-away that this is your climax is that the entire story has been leading up to this point. We've heard everything there is to hear about the birthmark, the possibility of getting rid of it, the danger of getting rid of it, etc., etc., and now we finally get to see the big moment. Ta-dah!
Suspense
Georgiana wakes up; she's all, "Poor Aylmer!", which we know can't be good.
Two things tip us off that there is something wrong. First is Aminadab's laughter, which we have a sneaking suspicion is not in the good spirit Aylmer believes it is intended. Second is Georgia's first words to her husband. Something is up, and we're on the edge of our seats to find out what. Also, there's only about a paragraph left for us to get to the conclusion.
Denouement
Georgiana is dying; we have the sneaking suspicion we knew this would happen all along.
Aylmer's dream sequence early in the story sets us up for this not-so-happy ending. There is a sense of falling action as we arrive at what, on some level, was an inevitable conclusion.
Conclusion
Georgiana is dead, and Aylmer is revealed to be a total fool.
Hawthorne's judgment, not ours, though he puts it in slightly kinder terms ("Had Aylmer reached a profounder wisdom, he need not thus have flung away [his] happiness"). There are big-time lessons to be learned on the nature of mortality, human nature, human imperfection, and the division of body and soul. Be sure to check out "What's Up with the Ending?" for a full plate of discussion.