To set up a free @facebook.com address, go to your Messages view and click the “Claim your Facebook email” link. Your email address will match your public username, for example:
If you don’t have a username yet, you can choose one when you create your email address. Once you set up your email address, people can email you using any traditional email system (e.g., Hotmail, Yahoo or Gmail), and the emails will be delivered to your Facebook Messages. When you send messages to external email addresses, the emails will be formatted to look like Facebook messages, including your name, your profile picture and your message.