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DOS
A Disc Operating System (or DOS) is an early example of a modern computer system designed to run on early versions of a Hard Drive. A main competitor to UNIX, DOS was co-developed by a number of people, most notably IBM (DOS), Apple (Apple-DOS) and Microsoft (MS-DOS). A console system, early-on it provided little features (intended to be run under a larger system) but as the Personal Computer market grew, more functionality was added.
Shells
A large number of DOS shells were made over the course of the DOS-boom (so to speak). These can be best compared to Linux's many Desktop Environments.
Current Status
The DOS days are over, and are currently only used by nerds. While no major company continues to provide DOS systems, some hackers have released and maintained their own DOS-clones, such as FreeDOS.
In the 32-bit versions of Windows (to this day), a DOS-based terminal can still be accessed through the file known as COMMAND.COM. This exists purely for compatibility reasons that Microsoft needs to keep in for whatever reason.
Emulation
DOSBox is the standard emulation tool for DOS. DOSBox is advanced enough to run Windows 3.1, and will run any DOS game you throw at it.