What is The Scarlet Letter?

It is both the title of a book and a punishment for crime that was used in the period that the book is set in. The main character of the book has been punished for a crime in that way. The punishment consisted of using a hot branding iron to brand the person's forehead with a letter identifying their crime. The burn scar from this branding when it healed turned bright red, thus the person wore a Scarlet Letter on their forehead for the rest of their life identifying them as a criminal and what their crime was.

The Scarlet Letter is a famous work of fiction published in 1850, and written by Nathaniel Hawthorne. If you are asking what the phrase "scarlet letter" means, it refers to the letter A for "adultery" which was worn for punishment by the woman who had committed that sin. The story takes place in an era when religion dominated the culture, and breaking one of the Commandments could cause you to be publicly shamed.