An SDK is a Software Developers' Kit. It includes all the software you need, such as the requisite header files and libraries. Some SDKs may contain pre-compiled libraries, and some (usually open source projects) may contain source code that you will need to compile into libraries yourself.
An API is an Application Programming Interface. It is, essentially, the header files declaring the data types and functions required to use the libraries. It also includes any instructions for using the library (e.g. call the init() function before calling any other function in the library).
So, an SDK contains the API. When posters refer to the API, they are (or at least should be) assuming the SDK is installed, and they are talking about how to use the library. When they refer to the SDK, they should be talking about all of the files that are needed to be downloaded onto your system to use the library.