"Unix" is a family of operating systems. A very long time ago, "UNIX" referred to something AT&T made, but this type of usage began to die out in the 1980's. Lately SCO is trying to revive this usage in order to confuse people who don't know better.
Conventionally, we call an OS "unix" if it implements certain features and works in certain ways.
There is now an official specification which anyone can try to comply with, defined by The Open Group (
www.unix.org). Most people, though, are happy referring to Solaris, IRIX, HP-UX, Linux, and maybe even AIX as being different flavors of unix operating systems. Currently, there is NO operating system anybody offers which is simply called plain old "UNIX." (and for reasons I won't get into, no such thing was ever marketed... the history is a little complex.)