Wikipedia on Unix:
Unix (officially trademarked as UNIX, sometimes also written as Unix with small caps) is a computer operating system originally developed in 1969 by a group of AT&T employees at Bell Labs, including Ken Thompson, Dennis Ritchie, Brian Kernighan, Douglas McIlroy, and Joe Ossanna. Today the term Unix is used to describe any operating system that conforms to Unix standards, meaning the core operating system operates the same as the original Unix operating system. Today's Unix systems are split into various branches, developed over time by AT&T as well as various commercial vendors and non-profit organizations.
As of 2007, the owner of the trademark is The Open Group, an industry standards consortium. Only systems fully compliant with and certified according to the Single UNIX Specification are qualified to use the trademark; others are called "Unix system-like" or "Unix-like".
.. on Unix-like:
*A Unix-like (sometimes shortened to nix to circumvent trademark issues) operating system is one that behaves in a manner similar to a Unix system, while not necessarily conforming to or being certified to any version of the Single UNIX Specification.
There is no standard for defining the term, and some difference of opinion is possible as to whether or not a certain OS is "Unix-like".
.. on Linux:
A Linux-based system is a modular Unix-like operating system. It derives much of its basic design from principles established in Unix during the 1970s and 1980s. Such a system uses a monolithic kernel, the Linux kernel, which handles process control, networking, and peripheral and file system access. [...]
Separate projects that interface with the kernel provide much of the system's higher-level functionality. The GNU userland is an important part of most Linux-based systems, [...]
.. on BSD (FreeBSD, NetBSD, OpenBSD):
Berkeley Software Distribution (BSD, sometimes called Berkeley Unix) is the UNIX operating system derivative developed and distributed by the Computer Systems Research Group of the University of California, Berkeley, from 1977 to 1995.
Historically, BSD has been considered a branch of UNIX — "BSD UNIX", because it shared the initial codebase and design with the original AT&T UNIX operating system. In the 1980s, BSD was widely adopted by vendors of workstation-class systems in the form of proprietary UNIX variants such as DEC ULTRIX and Sun Microsystems SunOS. This can be attributed to the ease with which it could be licensed, and the familiarity it found among the founders of many technology companies of this era. [...]
Today, the term of "BSD" is often non-specifically used to refer to any of these BSD descendants, e.g. FreeBSD, NetBSD or OpenBSD, which together form a branch of the family of Unix-like operating systems.
.. and on OS X 10.5:
Mac OS X v10.5 "Leopard" was released on October 26, 2007. It was called by Apple "the largest update of Mac OS X". [...] Leopard is an Open Brand UNIX 03 registered product on the Intel platform. It is also the first BSD-based OS to receive UNIX 03 certification.