A very simple way to describe them is that POSIX is one major "Unix standard". There are others, like XPG and SVID.
A quick search on Google pulled this up:
http://www.robelle.com/smugbook/posix.html
"Unix" as it was now ceases to be. The true decendant of the AT&T Unix source code is now held by Caldera, who bought out SCO. Check out
http://unix.about.com for lots of Unix history and facts. Many companies and groups put out their own Unix-like operating systems, for example: HP-UX, Solaris, *BSD, Caldera...
Linux was originally started as a Unix-like system that would run on checp PC hardware. Other alternatives include Hurd, Minix, the BSD family, and there's still other small ones running about.
Hope that answers a few questions...
