As a simplification, it's like this:
Compile something in Cygwin and you are compiling it for Cygwin.
Compile something in MinGW and you are compiling it for Windows.
About Cygwin
The purpose of Cygwin is to make porting Unix-based applications to Windows much easier, by emulating many of the small details that Unix-based operating systems provide, and are documented by the POSIX standards. Your application can use Unix feature such as pipes, Unix-style file and directory access, and so forth, and it can be compiled with Cygwin which will act as a compatibility layer around your application, so that many of those Unix-specific paradigms can continue to be used.
When you distribute your software, the recipient will need to run it along with the Cygwin run-time environment (provided by the file cygwin1.dll). You may distribute this with your software, but your software will have to comply with its open source license. It may even be the case that even just linking your software with it, but distributing the dll separately, may still require you to honor the open source license.
About MinGW
MinGW aims to simply be a Windows port of the GNU compiler tools, such as GCC, Make, Bash, and so on. It does not attempt to emulate or provide comprehensive compatibility with Unix, but instead it provides the minimum necessary environment to use GCC (the GNU compiler) and a small number of other tools on Windows. It does not have a Unix emulation layer like Cygwin, but as a result your application needs to specifically be programmed to be able to run in Windows, which may mean significant alteration if it was created to rely on being run in a standard Unix environment and uses Unix-specific features such as those mentioned earlier. By default, code compiled in MinGW's GCC will compile to a native Windows X86 target, including .exe and .dll files, though you could also cross-compile with the right settings, since you are basically using the GNU compiler tools suite.
MinGW is essentially an alternative to the Microsoft Visual C++ compiler and its associated linking/make tools. It may be possible in some cases to use MinGW to compile something that was intended for compiling with Microsoft Visual C++, with the right libraries and in some cases with other modifications.
MinGW includes some basic standard libraries for interacting with the Windows operating system, but as with the normal standard libraries included in the GNU compiler collection these don't impose licensing restrictions on software you have created.
For non-trivial software applications, making them cross-platform can be a considerable challenge unless you use a comprehensive cross-platform framework. At the time I wrote this the Qt framework was one of the most popular for this purpose, allowing the building of graphical applications that work across operating systems including Windows, but there are other options too. If you use such a framework from the start, you can not only reduce your headaches when it comes time to port to another platform but you can use the same graphical widgets - windows, menus and controls - across all platforms if you're writing a GUI app, and have them appear native to the user.