Unix is an operating system commonly used in internet servers, workstations and PCs by Solaris, Intel, HP etc.
Unix is a commercial operating system built by Bell Labs in 1969.
UNIX is a proprietary software, not truly open source, being copyrighted, only large companies like HP, IBM are allowed to use the Unix operating systems.
Different flavors of Unix have different cost structures
UNIX was ported to many other architectures.
Examples: OS X, Solaris, All Linux
Linux is an open source, free to use operating system widely used for computer hardware and software, game development, tablet PCs, mainframes etc.
Linux is an operating system originally developed by Finnish graduate student Linus Torvalds.
Linux is fully open source, there is no copyright issues involved here.
Linux can be freely distributed, downloaded freely, distributed through magazines, Books etc. There are priced versions for Linux also, but they are normally cheaper than Windows.
Linux is an operating system that is similar to Unix in its design. It is compliant with many of the same standards as Unix.
Examples: Ubuntu, Fedora, Red Hat, Debian, Android etc.