Posix is more as an OS, it is an "OS standard". You can imagine it as an imaginary OS, which actually doesn't exist, but it has a documentation. These papers are the "posix standard", defined by the IEEE, which is the big standard organization of the USA. The OSes implementing this specification are "Posix-compliant".
Government regulations prefer Posix-compliant solutions in their investments, thus being Posix-compliant has a significant financial advantage, particularly for the big IT companies of the USA.
The reward for an OS being fully posix compliant, that it is a guarantee that it will compile and run all Posix-compliant applications seamlessly.
Linux is the most well-known one. OSX, Solaris, NetBSD and Windows NT play here as well. Free- and OpenBSD are only "nearly" Posix-compliant. The posix-compliance of the WinNT is only a pseudo-solution to avoid this government regulation above.