Cygwin is is a Unix-like environment and command-line interface for Microsoft Windows.
Short for Minimalist GNU for Windows.
Mingw is a native software port of the GNU Compiler Collection (GCC) to Microsoft Windows, along with a set of freely distributable import libraries and header files for the Windows API. MinGW allows developers to create native Microsoft Windows applications.
You can run binaries generated with mingw without the cygwin environment, provided that all necessary libraries (DLLs) are present.