Hawthorne makes it clear to his readers that the birthmark is a symbol, mostly by telling us that it is a symbol. Check it out:
The crimson hand expressed the ineludible gripe in which mortality clutches the highest and purest of earthly mould, degrading them into kindred with the lowest, and even with the very brutes, like whom their visible frames return to dust. In this manner, selecting it as the symbol of his wife's liability to sin, sorrow, decay, and death. (8)
OK, so the birthmark represents Georgiana's humanity, which Hawthorne indicates is equivalent to representing her flaws. It is man's nature to be mortal and imperfect, he argues in this story – that's just what it means to be a human.
What does it mean, then, that Aylmer wants to remove the birthmark from his wife's face? On a literal level, he wants to take off what he considers to be an unattractive birthmark. But on a symbolic level, he wants to rid Georgiana of her flaws. He wants to make her perfect. Ironically, Aylmer succeeds – we'll talk about how he succeeds and where he fails in "What's Up with the Ending?"
For now, let's get back to this birthmark, and take a closer look at its physical appearance on Georgiana:
To explain this conversation it must be mentioned that in the centre of Georgiana's left cheek there was a singular mark, deeply interwoven, as it were, with the texture and substance of her face. In the usual state of her complexion — a healthy though delicate bloom — the mark wore a tint of deeper crimson, which imperfectly defined its shape amid the surrounding rosiness. When she blushed it gradually became more indistinct, and finally vanished amid the triumphant rush of blood that bathed the whole cheek with its brilliant glow. But if any shifting motion caused her to turn pale there was the mark again, a crimson stain upon the snow, in what Aylmer sometimes deemed an almost fearful distinctness. Its shape bore not a little similarity to the human hand. (7)
There's a lot going on in this paragraph. First, we note that the birthmark is "deeply interwoven" with Georgiana's countenance, which means symbolically that man's flaws and very much a part of his character and in fact cannot be separated out. It's also foreshadowing as to the story's ending; we know from the start that Aylmer is a fool to think he can rid her of something so deeply engrained in her face (literally) and character (symbolically).
Next, we note that the birthmark's visibility shifts with the changing color of Georgiana's face. Whether she's pale or flushed determines how much the birthmark shows. We also note that the birthmark is red – the color of blood, and the color of passion. One interpretation of this story is that the birthmark represents Georgiana's sexuality. Aylmer, uncomfortable with his wife's sexual power, wants to remove it to keep himself in control. There are lots of interesting articles to read along this vein, some more left field than others. For example, in "Speaking of the Unspeakable: Hawthorne's 'The Birthmark,'" Jules Zanger argues that the birthmark is actually about the menstrual cycle!
Moving on, let's talk about the shape of the birthmark. Hawthorne tells us that resembles a tiny hand. This is such an interesting line in a few different ways. First, the fact of a hand on Georgiana's face immediately makes us think of the hand of God – as though her maker touched her personally while crafting her apparently very beautiful face. But Hawthorne complicates the idea by specifying that it was the shape of a human hand – it is simultaneously the mark of Georgiana's humanity and mortality.
Looks like we've covered the major bases here – be sure to check out "What's Up with the Ending?" for more on the birthmark as a symbol.