SDK: A Software Development Kit (SDK) is a set of tools used to develop applications for a particular platform(Mac, Windows). An SDK typically contains a compiler, linker, and debugger. It may also contain libraries and documentation for APIs. SDKs also frequently include sample code and supporting technical notes or other supporting documentation to help developers. Often the SDK can be downloaded directly via internet many SDKs are provided for free.
IDE: Integrated development environment (IDE) contains certain controls and editor with the help of which designers can design the user interface and write code for each controls in the editor.
Example: Net beans, Visual studio.
Framework: Framework are classes provided by organizations which helps the programmers to perform complex task. Let's say earlier programmers used to code for database using lengthy procedures now that was time consuming so in order to ease their work, programmers will use a framework which will provide them certain classes and function which will help them to do the database connectivity or any other part regarding database within few steps.
Example:COCOA TOUCH (used for making i phone apps), .net framework, Java server faces
API: Application programming interface (API) is an interface between two software with the help of which they can communicate with each other. API is software to software interface (not a user interface) here's an example of API.
When you buy movie tickets and enter your credit card details then the movie ticket website uses an API to send your information to remote application that verifies whether your information is correct or not and once the information is correct the payment is deducted from your account and tickets are issued. At a time the user is only able to see one screen in this case the movie ticket website but behind the scene their are various application working together with the help of API.
Another example of API is facebook which uses google api to communicate with your gmail account to send friend request to your friends.