Neither Unix nor Linux is a single operating system. Rather, each are families of related operating systems.
"Unix" originally referred to an operating system developed in the 1960s by Bell Labs. The code at the core of this project eventually developed into many different lines, leading to most of the operating systems we know today. The various flavors of Linux are all descendants of this code, and so is MacOS X. In fact, almost every major OS in the world with the exception of the Windows family has some tie to Unix.
Today, "Unix" refers to a standard, the "Single Unix Specification." Any OS that meets this standard can call itself "Unix." All of them, of course, are descended from the original Unix project. The most widely known is Solaris, a Unix produced by Sun Microsystems.
There is nothing that prohibits a Linux operating system from meeting the Unix standard. That is, an OS could, theoretically, be both Linux *and* Unix. That said, there is no OS in existence that is both Linux and Unix.
"Linux" is not an OS at all. It is a kernel. A kernel is the code that lies at the heart of the operating system, and allows the OS to communicate with the hardware. Any operating system that incorporates the Linux kernel can be called a Linux OS.
Most Linux operating systems use the Linux kernel and take the other two major parts of the OS -- the shells and the applications -- from the GNU project. The GNU project is an attempt to make another OS descended from the original Unix. It is, at present, without a kernel that is ready for general release. This is why Linux and GNU go so well together -- one needs a kernel, the other needs everything but a kernel! Operating systems that include both GNU and Linux are more formally called GNU/Linux OSs. They include almost every widely known Linux distribution -- Ubuntu, Fedora, Gentoo, etc.
There is no reason that a Linux OS has to incorporate GNU components, though; it is simply the most common way to build a Linux OS.
To sum up, there is no answer to your question. Different Linux OSs can be radically different, with little in common. Similarly, there is no single OS called "Unix." Furthermore, at least in theory, the terms "Unix" and "Linux" are not mutually exclusive.
(If you like my answer, please select it as "Best Answer." If someone else's answer is better, please select their answer instead. But, please, select a best answer.)