Facebook launched its email-compatible messaging service, which assigned all account holders new facebook.com email addresses, in November 2010. Though expected in some corners to establish Facebook as a major player in the free email market—employees reportedly referred to the new messaging system as "the GMail killer" when it was launched—the Wall Street Journal notes that providers such as Google, Yahoo, and Microsoft continue to dominate the space.
What happens when external email comes to your Facebook.com account
Email sent to your facebook.com address will be routed to your Facebook "Messages" box. If the sender's email address is associated with the account of one of your Facebook friends (or one of their friends), the message will appear in your Inbox and you'll get the typical "talk bubble" notification at the top of the Facebook page.
If, however, Facebook can't match the sender's email address up with someone within two degrees of separation, it gets routed to a more obscure mailbox, "Other Messages," that many Facebook users are not even aware exists. Messages dropped in "Other Messages" do not generate a notification on Facebook. Some of these messages could be spam, but many could be from people whom Facebook doesn't realize are your legitimate contacts.
If you want to give facebook.com email a try, here's Facebook's own summary of how it works and how it looks and behaves differently from the email you're probably used to. This November 2010 LifeHacker article explains why that might not be a good decision for you. LifeHacker also offers instructions here on how to change your profile's default email back to what it was before.