Application is necessary to work with the API. Each application has its own set of id, a public key and a private key
SDK - ready to use implementation to interact with the API for the most popular platforms
REST methods allow you to obtain information about the current user, his friend list, etc.
Graph API - another HTTP-based solution for API
When working with the API some errors may happen, for example, passing an invalid parameter or expiration of user session
At common we do not have any limits for API calls but at some points we can limit API usage for certain applications and/or methods