Back in the day (colonial times, that is), law and religion were inseparable, like peanut butter and jelly. When a woman cheated on her husband—even if that husband had been missing for two years—she had to be punished by law, preferably in a creative way like having to wear a big red "A" on her dress.
When we begin The Scarlet Letter, Hester is carrying out her sentence by displaying her scarlet "A" (for adultery) for the community's enjoyment. They're digging it, even if one onlooker does say that the "A" should have been burned onto Hester's forehead. (Hey, she's being merciful: another onlooker thinks that the she should have been executed.)
Over time, though, the "A" becomes more than a symbol of adultery. It burns, glows, gets discarded, and taken back up again—almost as though it has a life of its own. And it's made so beautifully that the townspeople notice its artistry and craftsmanship before they remember what it means. Eventually, Hester is able to change the meaning of the letter and to earn the respect of those around her. The book's title puts the focus where it belongs: not on the individual (Hester) but on the letter that represents the community's response to her—and her response back.
And one more thing. Why do you think it's called The Scarlet Letter and not The Scarlet A? Does it just sound better?