Facebook provides a Software Development Kit (SDK) for using the Facebook APIs in various platforms and languages. It has a wealth of information, sample codes, and "How-tos" to help you get started quickly with integrating Facebook into your application.
All this information is available at https://developers.facebook.com/.
The Social Graph is a mapping of different people and how they are related to each other within a network. Facebook uses this term to refer to the Facebook platform, which was introduced in May 2007. Within the Facebook Social Graph context, every person, page, photo, or comment is a node that is connected to each other with the relations they share.