Unix (frequently typed as "UNIX"), today, is a catch-all name that may be used to describe any of a number of operating systems which are compliant to these closely-related standards: POSIX, The Single Unix Specification ("SuS") and ISO/IEC 9945.
While the original UNIX was developed at Bell Labs, the name today is applied to many of its direct descendants, as well as several operating systems and kernels which were inspired by it.
Note that an OS doesn't need to be fully Unix-compliant to be considered Unix-like.
The reverse is also true: a system can be Unix-compliant without being Unix-like.
Wikipedia has a brief discussion of Unix vs Unix-like.