Debian Operating System is an association of individuals who have made common cause to create a free operating system. This operating system is called Debian GNU/Linux, or simply Debian for short. Debian systems currently use the Linux kernel.
Linux is a completely free piece of software started by Linus Torvalds and supported by thousands of programmers worldwide. Of course, the thing that people want is application software: programs to help them get what they want to do done, from editing documents to running a business to playing games to writing more software.
Debian comes with over 8000 packages (precompiled software that is bundled up in a nice format for easy installation on your machine) - all of it free. It's a bit like a tower. At the base is the kernel. On top of that are all the basic tools.
Next is all the software that you run on the computer. At the top of the tower is Debian -- carefully organizing and fitting everything so it all works together.
Among versions of Linux, Debian is fairly unique, unlike most distributions it isn't backed by any for profit venture and instead is an all volunteer organization. There is no for profit company, or business plan behind Debian, it is a wide ranging project along the lines of GNU. Debian also provides the basis for a number of other Linux distributions, such as Lindows, Xandros, Libranet, and Knoppix.
The linchpin of Debian is a combination package management system and distribution network called forth by the mighty apt-get command. What makes this network so exceptional is that the dpkg-deb files which make up Debian packages contain information on their underlying dependencies so installing a package from the Debian network will call up all of its dependent packages.
As an example, installing a KDE application would make sure that KDE, X windows, and everything underneath was already installed on the system, and if the prerequisite packages weren't already installed than the packages would automatically be downloaded and installed from the Debian network.