From wiki:
Cygwin (pronounced /ˈsɪɡwɪn/,[2] SIG-win) is a Unix-like environment and command-line interface for Microsoft Windows. Cygwin provides native integration of Windows-based applications, data, and other system resources with applications, software tools, and data of the Unix-like environment. Thus it is possible to launch Windows applications from the Cygwin environment, as well as to use Cygwin tools and applications within the Windows operating context.
Cygwin consists of two parts: a Dynamic-link library (DLL) as an API compatibility layer providing a substantial part of the POSIX API functionality, and an extensive collection of software tools and applications that provide a Unix-like look and feel.
So, Cygwin is a set of tools which allows you to emulate a unix-like (or linux) environment on your windows machines.
The NDK is the Android Native Development Kit. It allows you to write parts of your application in native code (C/C++) and integrate them into your application. Your application still runs under the Dalvik VM but it can load shared objects creating using a cross compiler. The NDK contains all the necessary tools and build scripts to generate native code binaries. It's an advanced concept and one you should probably wait on until you fully understand the architecture.
The reason why it requires Cygwin (or some flavor of linux) is because it uses GnuMake and other linux tools such as awk or Nawk. These tools are not available (or are really hard to use) on windows platforms, hence the need for at least Cygwin (though I would advise you do yourself a favor and just install linux).