API means "Application Programming Interface". When you write a piece of software, there may be a human interface (like a GUI) which allows a user to interact with your software and an API interface, which allows a user program to interact with your software. The API is usually a collection of classes and functions which constitute a library.
In the context of GUI programming (like tkinter), "binding" usually means that you attach a function to be executed to a part of a GUI (like a button or a slider), so that if the user presses the button, the function is executed. One says that the function is bound to the button.