The simplest way to register or login with Facebook into your WordPress is by using the lightweight but powerful plugin called Facebook Login. Once activated you need to create an app and paste the id inside the settings page of the plugin.
That’s it! Now you will have a “Connect with Facebook” button on your login/registration page
The plugin also include avatars support. If you users connect with their Facebook account , the Facebook avatar will be shown instead of WordPress gravatar or mistery man.
In case you need extra features like integration with e-commerce plugins such a WooCommerce or EDD you can upgrade to the pro version. Along with the features mentioned above it also includes a powerful ajax registration / login form and it’s compatible with Popup Plugin to create ajax login popups.