GNU is an operating system that is developed by the GNU project. The GNU project is part of the Free Software movement and aims to build a free (as in freedom) operating system. The project consists in gathering people around the world to develop software that is freely available and can be modified and distributed without restrictions. To license GNU, Richard Stallman created the GPL (General Public License) that is widely used today to license all sorts of Free Software. This license grants to the developer and to the users four basic freedoms to use, share, modify and distribute software.
GNU stands for Gnu's Not Unix. Note that the word “Gnu” is part of the abbreviation GNU, this is called a “recursive acronym”. For some reason geeks like this kind of unorthodox playing around with words. Linux too follows the same pattern: Linux Is Not UniX.
Together with the Linux kernel, GNU form the GNU/Linux operating system that many call just 'Linux' for short. Linux is also licensed under GNU's GPL and thus it is Free Software. GNU and Linux could almost be considered as part of the same project but for historical reasons they were not developed by the same people. Those same historical reasons made the word 'Linux' much more popular than 'GNU' and today not many people relate GNU to the “Linux” operating system (note that I put Linux between brackets because it is not, strictly speaking, an OS, it's a kernel).
The GNU project started creating, one by one, all the software needed to run an entirely free OS. When they finally started creating a kernel they were very ambitious and decided to build a better version of the existing UNIX kernel, judging that they could improve performance, stability and security over the traditional monolithic kernel. Their kernel, called (the) HURD, eventually became usable after some decades of development but it never really matured. Much before they could finish the HURD, another developer created Linux and the spread of GNU/Linux operating systems immediately took off.
The developers of the HURD stated that they would not have started to develop it if Linux was ready to be used at the time. Since they started it, they consider just fair to continue and eventually, one day, create a kernel that is indeed much better than the ones we know today. If they are successful in doing so, we might see GNU/HURD (or maybe only GNU) operating systems. No matter what happens, GNU software is here to stay, be it with Linux or the HURD.
For more information check out the GNU website.