An IDE is an integrated development environment – a suped-up text editor with additional support for developing (such as forms designers, resource editors, etc), compiling and debugging applications. e.g Eclipse, Visual Studio.
A Library is a chunk of code that you can call from your own code, to help you do things more quickly/easily. For example, a Bitmap Processing library will provide facilities for loading and manipulating bitmap images, saving you having to write all that code for yourself. Typically a library will only offer one area of functionality (processing images or operating on zip files)
An API (application programming interface) is a term meaning the functions/methods in a library that you can call to ask it to do things for you – the interface to the library.
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 (e.g. extension code for using features of an operating system (Windows SDK), drawing 3D graphics via a particular system (DirectX SDK), writing add-ins to extend other applications (Office SDK), or writing code to make a device like an Arduino or a mobile phone do what you want). An SDK will still usually have a single focus.
A toolkit is like an SDK – it’s a group of tools (and often code libraries) that you can use to make it easier to access a device or system… Though perhaps with more focus on providing tools and applications than on just code libraries.
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). For example, .NET provides an application framework – it makes it easier to use most (if not all) of the disparate services you need (e.g. Windows, graphics, printing, communications, etc) to write a vast range of applications – so one “library” provides support for pretty much everything you need to do. Often a framework supplies a complete base on which you build your own code, rather than you building an application that consumes library code to do parts of its work.
There are of course many examples in the wild that won’t exactly match these descriptions though.