The Car Analogy
IDE: The MS Office of Programming. It's where you type your code, plus some added features to make you a happier programmer. (e.g. Eclipse, Netbeans). Car body: It's what you really touch, see and work on.
Library: A library is a collection of functions, often grouped into multiple program files, but packaged into a single archive file. This contains programs created by other folks, so that you don't have to reinvent the wheel. (e.g. junit.jar, log4j.jar). A library generally has a key role, but does all of its work behind the scenes, it doesn't have a GUI. Car's engine.
API: The library publisher's documentation. This is how you should use my library. (e.g. log4j API, junit API). Car's user manual - yes, cars do come with one too!
Kits
What is a kit? It's a collection of many related items that work together to provide a specific service. When someone says medicine kit, you get everything you need for an emergency: plasters, aspirin, gauze and antiseptic, etc.
SDK: McDonald's Happy Meal. You have everything you need (and don't need) boxed neatly: main course, drink, dessert and a bonus toy. An SDK is a bunch of different software components assembled into a package, such that they're "ready-for-action" right out of the box. It often includes multiple libraries and can, but may not necessarily include plugins, API documentation, even an IDE itself. (e.g. iOS Development Kit).
Toolkit: GUI. GUI. GUI. When you hear 'toolkit' in a programming context, it will often refer to a set of libraries intended for GUI development. Since toolkits are UI-centric, they often come with plugins (or standalone IDE's) that provide screen-painting utilities. (e.g. GWT)
Framework: While not the prevalent notion, a framework can be viewed as a kit. It also has a library (or a collection of libraries that work together) that provides a specific coding structure & pattern (thus the word, framework). (e.g. Spring Framework)