The MinGW from http://www.mingw.org/ does only support gcc 32 bit (host and target). The independent minGW-w64 project provides support for 64 bit, and also supports a much larger part of the Windows API. The MinGW-w64 project however does not provide official binary builds: These can be grabbed either from the personal build directories of the developers (the most popular being rubenvb), or from associated but independent projects like tdm-gcc or mingw-builds or msys2.

Source: http://wiki.qt.io/MinGW-64-bit

mingw and mingw32 are for creating executables for 32-bit windows systems. mingw64 is for creating 64-bit executables. Note: this doesn't have to do with what version you are running when you do the build, but what the target system is for the executable you are creating (the system on which you will be running the newly created executable).

Regarding MinGW and MinGW32, here's a snippet from
The MinGW Wiki

"The project's name changed from mingw32 to MinGW is to prevent the implication that MinGW will only works on 32 bit systems (as 64 and higher bit machines become more common, MinGW will evolve to work with them)."

To find out what version you have, go the the associated bin directory and do:

gcc --version

I'd recommend checking out minGW-w64, from here: http://mingw-w64.sourceforge.net/
That projects goal is to "deliver runtime, headers, and libs for developing 64 bit (x64), as well as 32 bit (x86), windows applications using gcc-4.6 or newer versions."

MinGW is based on GCC and binutils, along with other GNU software. GNU projects use so-called "GNU triplets" to specify architecture information. A triplet has this format:

machine-vendor-operatingsystem

The triplet was specified when the binaries were built.

MinGW-w64 is using w64 as the vendor field, and mingw32 as the operating system. Of course, mingw32 is not an operating system, but it is treated as one by the GCC and binutils build system and source code.

So these long names for the binaries are an artifact of the build system that GCC and binutils are using. All of these binaries are installed in the same directory, and so binaries that target different architectures (in this case 32-bit vs 64-bit) need to have different file names. The easiest way to do that is to prefix the binaries with the GNU triplet that describes those architectures.

mingw-w64 is fork of the mingw project.

Generally the mingw-w64 headers are much more complete, they include more recent additions from newer Windows (new SDK headers, function declarations, types, constants, macros), are ready for building 64-bit targets (when used with gcc capable of that).

Note of course that you cannot mix mingw's gcc with mingw-w64 headers or vice versa. You have to use complete toolchain from one or the other. I don't know how easy it is to switch toolchain in the devc++ (I don't use it).

MinGW is a GCC-port for Windows. Not all of the Windows API is supported (but for many programs the supported stuff is sufficient) and it´s only for 32bit-Programs (which often can run on 64bit-Windows too, but some can´t, and you can´t compile them as 64bit).

MinGW-w64 is a improved version which supports both 32bit and 64bit, and some more of the WinAPI (still not all, because thats much work, but more than MinGW).

MinGW-w64 only provides their source code, but no binaries to "just use" the compiler.

MinGW-builds is a somewhat separate project to provide binaries in the most useful configurations. To get a specialized build of MinGW-w64, manual compiling is still possible.

Using the MinGW-builds self-installer is the easiest way, if nothing unusual is needed. Also see here for help with the self-installer.

W32 and W64 are two different architectures, so you distinguish by compiler name. The debian mingw-w64 calls them i686-w64-mingw32-gcc for Win32 and x86_64-w64-mingw32-gcc for Win64.