The first efforts at developing a multi-user, multi-tasking operating system were begun in the 1960's in a development project called MULTICS. While working for Bell Telephone Laboratories in 1969 and 1970, Ken Thompson and Dennis Ritchie began to develop their own single-user, multi-tasking small operating system and they chose the name UNIX. Their initial goal was simply to operate their DEC PDP machines more effectively. In 1971, UNIX became multi-user and multi-tasking, but it was still just being developed by a small group of programmers who were trying to take advantage of the machines they had at hand. (In other words, this operating system that they were developing did not run on any machine made by Bell!)
In 1973, Dennis Ritchie rewrote the UNIX operating system in C (a language he had developed.) And in 1975, the portability of the C programming language was used to "port" UNIX to a wide variety of hardware platforms. For legal reasons, Bell Labs was not able to market UNIX in the 1970's, though they did share this operating system with many universities - most notably UC-Berkeley. This led to some of the variations in UNIX which we see today. After the divestiture of the Bell System, their parent company, AT&T, became much more interested in marketing a commercial version of UNIX. And today we see that many companies have now licensed their own version:
AT&T's System V,
Versions of System V such as SCO's Xenix and IBM's AIX
Berkeley's UNIX (called "BSD" for "Berkeley System Development"), Versions of Berkeley UNIX such as Sun Microsystem's SunOS, DEC's Ultrix and Carnegie Mellon University's Mach
(used on the NEXT).