A simple explanation of what is MinGW
To compile C program you need a C implementation for your specific computer.
C implementations consist, basically, of a compiler (its preprocesser and headers) and a library (the ready-made executable code).
On a computer with Windows installed, the library that contains most ready-made executable code is not compatible with gcc compiler ... so to use this compiler in Windows you need a different library: that's where MinGW enters. MinGW provides, among other things, the library(ies) needed for making a C implementation together with gcc.
The Windows library and MSVC together make a different implementation.
MinGW is a suite of development tools that contains GCC (among others), and GCC is a C compiler within that suite.
MinGW is a complete GCC toolchain (including half a dozen frontends, such as C, C++, Ada, Go, and whatnot) for the Windows platform which compiles for and links to the Windows OS component C Runtime Library in msvcrt.dll. Rather it tries to be minimal (hence the name).
This means, unlike Cygwin, MinGW does not attempt to offer a complete POSIX layer on top of Windows, but on the other hand it does not require you to link with a special compatibility library.
It therefore also does not have any GPL-license implications for the programs you write (notable exception: profiling libraries, but you will not normally distribute those so that does not matter).
The newer MinGW-w64 comes with a roughly 99% complete Windows API binding (excluding ATL and such) including x64 support and experimental ARM implementations. You may occasionally find some exotic constant undefined, but for what 99% of the people use 99% of the time, it just works perfectly well.
You can also use the bigger part of what's in POSIX, as long as it is implemented in some form under Windows. The one major POSIX thing that does not work with MinGW is fork, simply because there is no such thing under Windows (Cygwin goes through a lot of pain to implement it).
There are a few other minor things, but all in all, most things kind of work anyway.
So, in a very very simplified sentence: MinGW(-w64) is a "no-frills compiler thingie" that lets you write native binary executables for Windows, not only in C and C++, but also other languages.