An Application Programming Interface or API, classifies the classes, methods, functions and variables that one application will need to call in order to tasks. APIs can be procedural or object oriented. The procedural API helps you to call functions to carry out tasks. And the object-oriented API creates class instantiation and then calls methods on resulting objects. The object-oriented API is much preferable as they lead to better organized code.
The connector is a piece of software that allows your applications to connect to the MySQL database server. MySQL provides connectors for variety of languages including PHP. In a dynamic database driven web applications, you need to write php code to perform a few tasks including connecting to the database server, querying the database and other database related functions. The Connector is a kind of software that allows your application to connect to a database server.