What is Linux?
Linux is an Open-Source computer Operating System originally developed by Linus Benedict Torvalds after whom it is named. At the heart of a Linux OS is the Linux Kernel. Torvalds wrote the first Linux kernel and still heads the its development.
Linux was modelled on the Unix operating system, which is considered by the Computer Industry to be a very robust and powerful OS with inherent security and multi-user capabilities. Linux is a Unix implementation and regardless of what the copyright owners of Unix say, it indeed is Unix.
Linux is available in the form of many distribution and hence many different flavors, some of which are Red Hat Linux (now Fedora project), Mandrake Linux, Debian Linux, Suse Linux (now taken over by Novell), Free BSD and many more.
Most flavours of Linux are shipped with the latest GNU softwares.
For further details, visit http://www.linux.org

What is Unix?
Unix is an operating system originally developed by Ken Thomson. It later grew to become a major force in the Industry, especially in networking application. Most of the Internet is powered by Unix and Linux based machines.
Due to it's open architecture, many flavours of Unix came into existance, due to which, later on, inter-operability problems started to arise out of the absence of a uniform interface between applications, programs and network protocols. To control this chaos, a body called POSIX came into being and laid down certain guidelines for all Unices to comply with.
Definition: Any system that complies with POSIX is Unix.
Hence Linux is Unix.
For further details, visit http://www.unix.org, http://standards.ieee.org/regauth/posix and http://www.pasc.org

What is Open-Source?
Open-Source necessarily means that the source code (and not just the executable binary machine code) of a software system is made available to the recieving party. The open-source system may (according to aggreement) be subject to re-use, modification or any other use by the recieving party. That is, the source code (and hence the right to know how the software is made) becomes available to the recieving party.
The recieving party may be the private party who ordered for the software to be made or the general public (depending on the license aggreement)
For further details, visit http://www.opensource.org

What is Free Software?
Free software is that software which the users/recipients can use without paying compensation in any form to the developer for development of the software.
The recipient, however may have to bear the cost of the medium through which the software was reached (such as the CD) and the commission of the middle-men (vendors, couriers, etc.) who brought it to the recipient.
For further details, visit http://www.fsf.org

What is GNU?
GNU is a recursive acronym for "GNU is Not Unix"
GNU is a foundation originally initiated by Richard Stallman of the GNU/Linux fame.
Linux forms the basic OS and GNU provides all the software on top of the Linux OS in order to make it useful to the end users like you and me.
For further details, visit http://www.gnu.org