DOS
DOS was an operating system originally written by Tim Paterson, heavily based on CP/M. It was later bundled with IBM PCs and their clones.
QDOS
In 1979, Microsoft hired Tim Paterson to help Bob O'Rear port their BASIC interpreter across to the 8086, as BASIC-86. It had its own filesystem, allowing each file to have up to eight characters in its filename proper and three more in its extension.
Seattle Computer Products bundled BASIC-86 with their 8086-based computer kit, but they really wanted to bundle a fully fledged operating system such as Digital Research's 8086-based port of CP/M. The only problem was it wasn't finished yet.
Tired of waiting, in 1980, Seattle Computer Products hired Paterson to write a rival operating system that combined the functionality of CP/M (right down to the API level) with Microsoft's filesystem. He completed it later that year, naming it QDOS, short for the Quick and Dirty Operating System.
MS-DOS
When IBM later made their own 8088-based personal computer, the rather generically named IBM PC, they failed to reach an agreement with Digital Research for CP/M. As an alternative, they licensed QDOS from Microsoft, who in turn licensed it from Seattle Computer Products. Microsoft hired Paterson to help O'Rear port DOS to the new 8088-based machine. After implementing a lot of changes demanded by IBM, they finally completed the new port of DOS the month before IBM publicly unveiled their PC in August 1981. Microsoft outright bought DOS from Seattle Computer Products, and renamed it MS-DOS, the Microsoft Disk Operating System. Meanwhile, IBM called it IBM PC DOS.
Operating systems: DOS | IRIX | TOS
DOS: Impulse Tracker | Scream Tracker 3