Let's see if I remember my history.
Unix was a proprietary (non-free) operating system. The GNU project set out to create a free (open source) clone of Unix, but they didn't have a kernel ready.
Enter Linus Torvalds, a Finnish programmer. He wrote the Linux kernel as a hobby, releasing it under the GPL (general public license). The Linux kernel was similar to Unix, but not exactly the same. This kernel, coupled with the GNU applications, forms the basis for the many distributions we know today, generically referred to as 'Linux'.
So, short answer: Linux is open source, Unix is closed.

Unix is primarily based on ATT's code (System V and such). It was originally created for the DEC PDP machines and now runs primarily midrange hardware from Sun, IBM, and HP.

Linux was created with more emphasis on microcomputers (Intel x86, for example) and hobbyist rather than corporate use. It's Unix like (as mentioned in the other comments), but is not Unix.

I believe the current arbiter of what is actually Unix is Novell, IIRC.

Both systems generally conform to the Open/POSIX standards (not Open like Open Source, but Open like documented standards).