Is Linux a Unix?
That depends on what you mean by “Unix”, and by “Linux”.
UNIX is a registered trade mark of The Open Group. The trade mark has had an eventful history, and it's not completely clear that it's not genericized due to the widespread usage of “Unix” refering to Unix-like systems (see below). Currently the Open Group grants use of the trade mark to any system that passes a Single UNIX certification. See also Why is there a * When There is Mention of Unix Throughout the Internet?.
Unix is an operating system that was born in 1969 at Bell Labs. Various companies sold, and still sell, code derived from this original system, for example AIX, HP-UX, Solaris. See also Evolution of Operating systems from Unix.
There are many systems that are Unix-like, in that they offer similar interfaces to programmers, users and administrators. The oldest production system is the Berkeley Software Distribution, which gradually evolved from Unix-based (i.e. containing code derived from the original implementation) to Unix-like (i.e. having a similar interface). There are many BSD-based or BSD-derived operating systems: FreeBSD, NetBSD, OpenBSD, Mac OS X, etc. Other examples include OSF/1 (now discontinued, it was a commercial Unix-like non-Unix-based system), Minix (originally a toy Unix-like operating system used as a teaching tool, now a production embedded Unix-like system), and most famously Linux.
Strictly speaking, Linux is an operating system kernel that is designed like Unix's kernel.
Linux is most commonly used as a name of Unix-like operating systems that use Linux as their kernel. As many of the tools outside the kernel are part of the GNU project, such systems are often known as GNU/Linux. All major Linux distributions consist of GNU/Linux and other software.
There are Linux-based Unix-like systems that don't use many GNU tools, especially in the embedded world, but I don't think any of them does away with GNU development tools, in particular GCC.
There are operating systems that have Linux as their kernel but are not Unix-like. The most well-known is Android, which doesn't have a Unix-like user experience (though you can install a Unix-like command line) or administrator experience or (mostly) programmer experience (“native” Android programs use an API that is completely different from Unix).
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.