A set of GNU utilities, compilers, and shells to make Microsoft Windows work like a GNU or Unix system. Very cool. Very nice.
Cygwin is a part of a commercial package that Red Hat develops - called GNUPro, which is pretty expensive (roughly in the $10000 class).
Cygwin itself is an emulation layer for the GNU C library calls to the Windows API. With it, you can relatively easily port programs from UNIX-like systems to Windows environments. These programs rely on a special dll (Dynamically Linked Library) file for 'translating' library calls.
In fact, with Cygwin you can also develop applications specifically for a native Windows environment with the aid of some libraries and header files developed for the MinGW project.
Cygwin is actively developed and it has a very neat installer that downloads the appropriate packages from a mirror server near you. Practically hassle-free.
And yes, with Cygwin you can now run X on Windows.
Recently even Gnome is available for Cygwin. Which means you can run a complete UNIX-like desktop inside Windows.
Cygwin is published under the GPL by Cygnus, now a part of Red Hat.