This contrib module provides integration between Facebook and Moodle.
What this module does:

• Provides a new profile field to hold the users Facebook ID & display the users Facebook picture
• Authentication plugin to authenticate users against Facebook
• Activity - Facebook stream box (Facebook chat)

This module does not attempt to sync any data (such as name and picture) with Facebook, rather it uses the Facebook servers (as the Facebook API suggests).

I've spent a bit of time working on this so it should be fairly complete. Further installation instructions are in the included README.txt