A software development kit can be many things, but generally, it’s a package of source files, usually that define new data types and subroutines and classes to make programming for a particular technology easier.
For example, to write programs for Windows using C++, you 
need information on what Windows functions and data structures are available, then you’d need to write C++ code to those specifications so that you can call those methods and you’d need to write code that defines the data structures that those methods expect you to pass to them.
Fortunately, Microsoft has provided a Windows SDK that has all of this already written. Now, all you need to do is reference those files from your own source files, then those complex data structures and function calls are available for your program to code against, without you having to reinvent the wheel.
There are SDKs for plenty of other technologies as well, and not just for operating system level stuff. There are SDKs for writing games. For example, to write “Games for Windows” (a generic sounding label for a branded technology), Zune, and
XBox 360, you can download the XNA Framework, which is an SDK for writing those types of games. That SDK greatly simplifies writing games for those three platforms.
There are thousands of SDKs for multiple platforms for multiple products for multiple technologies. Some SDK include more than just source code. Some include already compiled libraries and/or end user (for programmers) utilities.