POSIX
POSIX is, strictly speaking, not an operating system but a standard for Unix-style operating systems. An independent standards body defined and maintains the standard.
For more information on POSIX, see the IEEE standards Web site.
Linux
Linux is a POSIX-compliant implementation of Unix that is distributed as open source using the GNU licensing model.
The base Linux distribution does not directly address the real-time operations necessary for many embedded applications. However, there are efforts under way to support real-time facilities in Linux. A good source of information is the Real-Time Linux Web site.
Proprietary operating systems
There are many different proprietary operating systems designed for embedded applications. You can find pointers to some of them at this site's links page.