What does the scarlet letter represent for Hester?

The scarlet letter is a public sign of humiliation for Hester, a sign that she's a "fallen woman," an adulteress, someone who doesn't really belong in this God-fearing Puritan community. At that time, the prevailing double standard held that women, not men, were responsible for adulterous liaisons. And so it's Hester who's subject to such degrading, unjust treatment for a moral transgression in which it takes two to tango.
Over time, however, Hester manages to forge a new identity for herself, and the scarlet letter plays an important part in this. The letter, though still a very potent symbol of sin and public humiliation, is also conversely, a sign of relative freedom. As she is now a social outcast, Hester is no longer subject to the stifling restrictions of Puritan life. This gives her the opportunity to think about a new life for herself and her daughter, something that would've been unthinkable without the scarlet letter.