Real Mode
Real Mode is a 16 bits Operating mode in which the X86 CPU runs when it boots. That mode is mainly for backward compatibility and provide very few help for the modern developer (no memory protection, only 1MB of addressable memory, no virtual memory support). BIOS and DOS are typically real-mode stuff. All the rest you know (Windows, Linux, DukeNukem3D, zsnes, dos4gw, djgpp ...) are Protected Mode OS/applications/dosextenders respectively.
See Also
External Links
http://www.osdever.net/tutorials/rm_addressing.php?the_id=50