A chunk of code that you can call from your own code, to help you do things more quickly/easily
API
The functions/methods in a library that you can call to ask it to do things for you - the interface to the library.
SDK
An SDK (software development kit) is a library or group of libraries (often with extra tool applications, data files and sample code) that aid you in developing code that uses a particular system.
Examples:
Windows SDK
DirectX SDK
Office SDK
Framework
A framework is a big library or group of libraries that provides many services (rather than perhaps only one focussed ability as most libraries/SDKs do).