What is the difference between the next terms, it can help a lot in interviews and general understanding.
Framework
Some predefined architecture that a developer has chosen and which dictates how the application will be written. It usually already includes many concepts which helps the developer to concentrate on the domain of the application instead of the plumbing. This plumbing is provided by the framework. For example the .NET framework provides out-of-the-box tools that would allow you to talk to web servers, without even knowing the internals of the TCP/IP protocol (actually it helps knowing the internals but you get the point).
Library
A reusable compiled unit that can be redistributed and reused across various projects. Well not necessary compiled in case of dynamic languages.
IDE
It's the development environment where you create the other three parts (usually text editor), it might also include compiler and the possibility to execute, debug and see the output of the program in order to speed up the development process.
API
Application Programming Interface. This could mean many things but usually it is a set of functions given to the disposition of the developer and which perform specific tasks and work only in a specific context.