What does it mean to have a POSIX-compliant operating system?

All you want/need to know about POSIX, and POSIX-compliant OSes in particular.

POSIX (Portable Operating System Interface for Unix) is simply a set of standards that define how to develop programs for UNIX (and its variants). Being POSIX-compliant for an OS means that it supports those standards (e.g., APIs), and thus can either natively run UNIX programs, or at least porting an application from UNIX to the target OS is easy/easier than if it did not support POSIX. Of course, the level of compliance is not necessarily 100% and can vary (e.g., not all features are supported or may be implemented differently).

It simply means that the operating system complies with the POSIX (Portable Operating System Interface for Unix) family of IEEE standards. Five minutes on Google will give you a list of compliant OS' and that actual content of the standards.