So, there are lots of different versions of Unix out there: HP-UX, AIX, BSD, etc. Linux is considered a Unix clone rather than an implementation of Unix. Are all the "real" Unices actual descendants of the original? If not, what separates Linux from Unix?
That depends on what you mean by “Unix”, and by “Linux”.
For all intents and purposes, a typical modern Linux distribution (Ubuntu, Debian, Red Hat, Fedora, Slackware, etc) is a Unix, but strictly speaking, no system can claim to be Unix without being certified, so instead people say they are Unix-like. They are inspired by Unix, and carry on its culture.
This also applies to BSD systems.
Mac OS X is certified Unix, so it's Unix both in name and indeed. (and it's actually based on BSD).
It should be noted that since Linux itself is just a kernel, it can be used to build non-unix-like systems (such as Android).
Linux is more 'Unix-like' so yes simplistically you could call it a clone, the same is true for BSDs (although admittedly BSDs could be considered closer to Unix than Linux).
The main thing that gives Linux the Unix-like title is the fact that it is nearly fully compliant w/ POSIX (Portable Operating System Interface [for Unix]) standards that have built up over time.
The other key thing, is the inheritance of code etc, as demonstrated on Wikipedia, Linux does not actually originate from Unix sources, instead it is pretty much built from scratch (this is disputed however).
So essentially, the main thing that separates Unix from Linux is the ancestry and the standards that it meets.
The choosen answer explains it quite exhaustively, but you would have to watch a whole documentary to know all the details between the UNIX initial developement, and how Linux went out.
First you have to consider that Linux is the kernel, which was made by linus and other programmers. He chose to release it with the GPL, which by the time was a license made by the FSF, (RMS and other folks), who were also developping their own kernel and GNU.
What we usually call Linux is, at first, the kernel, plus all other tools originally coming from the GNU project. Those two project are historically different, because at the time, the kernel from the FSF folks was abandonned to favor the Linux kernel, which was much better.
I remember there is a documentary, you should really watch it, I find it important for your programming culture.
Another thing you have to consider when thinking about "*nix", is also everying involving POSIX and other architecture-standardised stuff. It's subject of OS design/research, but it defines precisely how the involved OSes works, and is crucial when you have a kernel working well with its tools.
Linux is a Unix-Like Operating System developed by Linus Torvalds and thousands of others.
Linux is the most prominent example of a "real" Unix OS. It runs on anything and supports way more hardware than BSD or OS X. An interesting quote I found when I was comparing BSD and Linux:
BSD is more like a Unix OS than Linux is. Also notable is that Apple makes use of BSD and Linux components. Apple Uses APT from Debian and Ubuntu on the iOS and OS X platforms. And it is based on BSD. (The kernel though is Darwin, which is it's own kernel. Beastie the platypus is the Darwin mascot because he is a mix between Beastie from BSD and a Platypus.)
If you want a "real" Unix operating system (One that runs on anything and supports lots of hardware), try Linux.
I'm a long-time Linux User, having used it off and on from the 90's to early 2000's and then quit using it for awhile but started using it again around mid 2012 as my permanent OS so I can recommend it to anyone who wants to try something other than Windoze.