GCC comes with a compiler runtime library (libgcc) which it uses for (among other things) providing a low-level OS abstraction for multithreading related functionality in the languages it supports. The most relevant example is libstdc++'s C++11 <thread>, <mutex>, and <future>, which do not have a complete implementation when GCC is built with its internal Win32 threading model. MinGW-w64 provides a winpthreads (a pthreads implementation on top of the Win32 multithreading API) which GCC can then link in to enable all the fancy features.
I must stress this option does not forbid you to write any code you want (it has absolutely NO influence on what API you can call in your code). It only reflects what GCC's runtime libraries (libgcc/libstdc++/...) use for their functionality. The caveat quoted by @James has nothing to do with GCC's internal threading model, but rather with Microsoft's CRT implementation.
To summarize:
posix: enable C++11/C11 multithreading features. Makes libgcc depend on libwinpthreads, so that even if you don't directly call pthreads API, you'll be distributing the winpthreads DLL. There's nothing wrong with distributing one more DLL with your application.win32: No C++11 multithreading features.Neither have influence on any user code calling Win32 APIs or pthreads APIs. You can always use both.