Technically "Linux" (what Torvald Linus wrote) is just the kernel, a kernel which is used together with commands and tools from the GNU-project (by Ricard Stallman) to make a complete Unix-like operating-system. Some feel that this combination should be called "GNU/Linux", but most just call the whole thing simply "Linux" - meaning both the Linux-kernel and GNU commands & tools.
While the kernel is the most important part of an operating-system, it alone does not make an opertaing system. It needs other commands and tools to deal with things like copying and removing files, making directories, editing simple files, adding/deleting users/groups and monitoring/controlling processes. For Linux - or rather GNU/Linux - systems, these commands and tools are strictly speaking part of the GNU-project, and not part of "Linux".
GNU - GNU is not Unix - was project started by Ricard Stallman of the Free Software Foundation (the people behind the GPL - GNU Public Licence) to create a free, open-source Unix-like operating system. They made lots of packagaes - compileres (gcc), editors, commands (cp, mv, rm, ...), shells and all sort of stuff - but what GNU missed, was the actual kernel... the part that would communicate with the computer hardware. Stallman wanted to go with Hurd - a kernel based on the Mack micro-kernel - which not only was modern and revolutionary (as far as kernels goes), but also introduced lots of interesting new features and possibilites. However, before the Hurd-kernel was mature enough to be used, along came the Finish student Linus Torvalds with his Linux-kernel - a monolitic old-fashion kernel (as opposed to a more modern micro-kernel) - and provided the "missing piece" in the GNU-project... and - to the (understandable) annoyance of Stallmann - most people started refering to the whole thing (Linux *and* GNU) as simply "Linux".
Debian is one of many "Linux distributions" or "distros". A distro is something a vendor makes, by putting the Linux/GNU operating system - together with a selection of lots and lots of (usually free and open) programs to run under the operating system - on a DVD or CD-ROM. Typically the vendor will make sure that it works and different parts (programs) plays well with each other. The installation-program can vary much between distros. Software are usually provided as pre-compiled "binary-packages", though different distros may use different package formats and tools to add/remove/manage packages - especally by making sure packages a package depends on are also installed. A package for one distro, will often not work with another distro. Traditionally Unix-software were distributed as source-code, and compiled on-site... this is still an option, but with binary packages you're usually spared this. Different distros will vary in regards to which packages are available, which are "default" and with which are installed on a "typical system" (especially which WindowManager/Desktop Enviroment is the default - which also dictates many of the other "standard" programs, like mail-clients and office packages). A great thing, is that daemnons for various internet-services - like web-server, mail-server, file-server and databases - are among the packages, making it easy to set-up a server at home. Debian is as I said *one* of many distros - others are RedHat, Mandriva, Fedora, Suse, SlackWare, Ubuntu, Gentoo and many more. http://distrowatch.com gives you a list, together with some non-GNU(Linux open-source OSes.
Debian is the distro ran by the FSF/GNU-project itself. It's two primary goal is to always fully support many different acitechtures (not just PC and intel-processors, but many other platforms like the Alpha-processor) and to offer a very large number of packages. Sadly, as they almost must make sure it work on all platforms, they often lag a bit behind. Software comes in deb-packages (which they invented, as opposed to the rpm-packages invented by RedHat and also used by Mandriva). Debian is also the "base" of the wastly popular Ubuntu-family of distros, as Ubuntu is based on a Debian-system, but where the release-cycle of new packages has been shortened a lot (as Ubuntu is just for the PC-platform). Ubuntu also uses deb-packages, but it's best to not mix Ubuntu and Debian-packages too much.