What is the difference between MinGW, MinGW-w64 and MinGW-builds?

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.