Oh, that is a loaded question, right up there with differences among the christian religions, or explaining chocolate and strawberry ice cream.
Unix was developed by a group of AT&T employees at Bell Labs in 1969. From there different groups have done different things with it.... see WikiPedia for details.
Linux was created by a Computer Sciences student Linus Torvalds in 1991. His release of a kernel (core code) happened in the early days of the internet, which allowed talented programmers to look into it, and write things against it (meaning, link into the core, such as a device driver for a keyboard that the kernel could process), and Linux was born. See WikiPedia too.
I am of the opinion that Linux and Unix are quite related, but there are some important differences. To end users, they may feel the same, but notice that file structures are different (is that config file in /usr/etc, or /etc) and differences in what sorts of software is available. You may say that they are like flavors of ice cream... basically the same, with different tastes.
Mac OS X is a flavor of Unix. Apple has a special GUI interface running over the Unix. Apple might be mint chocolate chip ice cream.
Windows is not a flavor of ice cream. It is more like a cookie. Tastes good for some, but leaves lots of crumbs all over the place.
Both Linux and Unix have different codesets within them, called distributions (or distros). Linux has a variety of them, mostly based on the RedHat, Debian, or Slackware model. Unix also has different sets, such as BSD (Berkeley), Solaris, and IBM AIX. Also, Apple OSX is Unix.
What makes Unix and Linux special is that code compilers are included in the operating system, and you can either locate a program binary (such as Firefox to look at web pages), or find the source code, and compile it all yourself. There is even a Linux out there called LFS - Linux From Scratch, where you get a small boot loader thing, and compile from the ground up. Usually a training exercise -- few of us Linux folk compile from the ground up.