The quick answer is GNU's Not Unix. It was started by Richard Stallman for purposes that shall be revealed...
In the beginning, the plan was pretty clear, namely to create a "free" operating system rather like Unix, only available under a free license. The point of this wasn't primarily for it to be "free of cost," but rather for it to be "freely usable," so that people could examine, understand, and change the source code as desired.
The project began with a text editor, Emacs, and quickly progressed to include the creation of a C compiler, GCC, as that proved to be a crucial dependancy. Text processing utilities "in Unix style" were created. One of the later goals being the creation of an operating system kernel, which eventually became the Hurd project.
The FSF, or Free Software Foundation, was founded as a corporate structure to provide such things as:
An organization to encourage dissemination of GNU software;
A corporate framework to which to attach legal support for the GPL licensing of GNU software;
In its role as a tax exempt charity, a nexus to attract financial support so that grateful users of GNU software could sponsor production of more such software;
A nexus from which to fund the development of GNU software.
Ambiguity now arises; there are various things here that are identifiable that bear some sort of relationship to one another but not a clear and unambiguous relationship.
There is all sorts of software licensed under the GPL; the use of the license does not necessarily make it part of the "GNU Project."
For instance, the GNOME Project, creating libraries for GUIed applications is associated with the FSF and "GNU." But the KDE Project, also using the GPL and LGPL, is apparently not part of the "GNU Project."
Officially, "GNU Software" is is software that is released under the auspices of the GNU Project. Unfortunately, this categorization is not terribly precise; apparently "under the auspices of the GNU Project" represents a somewhat Zen-like "GNU Nature" whereby those that are "officially" members of the GNU Project determine what software, under what licenses represent part of their project.
At one time, it was thought that Hurd would be the "GNU Kernel," a necessary component of a "GNU System."
But since that time, what with the rather slow progress of Hurd development, and the rather more rapid progress of development of the Linux kernel, the FSF and RMS have seen fit to consider the combination of Linux and "GNU Tools" to represent a "GNU System."
It more or less appears that for something to be "officially" part of the GNU Project, it needs to be blessed by one of those that are "officially" part of the Free Software Foundation, and more particularly, by someone who happens to have the initials RMS .
I'm not sure that I entirely like that.