IBM PC-DOS was one of the three major operating systems that dominated the personal computer market from about 1985 to 1995. The original 1981 arrangement between IBM and Microsoft was that Microsoft would provide the base product (at that time a rather primitive clone of the market-leading CP/M-86), and that both firms would work on developing different parts of it into a more powerful and robust system, and then share the resultant code. MS-DOS and PC-DOS were to be marketed separately: IBM selling to itself for the IBM PC, and Microsoft selling to the open market.
For many years MS-DOS and PC-DOS were so nearly identical that most people tended to confuse them, and a program written for either one could be virtually guaranteed to work equally well on the other. It is commonplace to see people write "MS-DOS was the operating system for the original IBM PC", for example - which is quite wrong: the IBM PC always shipped with PC-DOS in the traditional IBM blue wrapper. Other companies also produced their own branded DOS versions: mostly just badge engineered MS-DOS under licence from Microsoft, but the MS-DOS based Compaq DOS included significant customisation, notably with version 3.31 which was clearly the most advanced DOS on the market for quite some time (if we ignore the CP/M-86 and DR-DOS family of products).
For DOS versions 1 through 5, the differences between MS-DOS and PC-DOS remained trivial. After the release of DOS 5.0, however, IBM and Microsoft - until then the closest of allies - had a serious falling out. The primary issue was the future of more advanced operating systems - Microsoft favouring Windows because it was easier to market and they owned 100% of it, IBM favouring the much more ambitious and technically sophisticated joint IBM/Microsoft OS/2 project - but the ramifications for the IBM-Microsoft business relationship were broader. From this time on, MS-DOS and PC-DOS would diverge, and for the first time, IBM would start actively marketing PC-DOS to other computer manufacturers and to the public at large.
IBM PSP (their Personal Software Products arm) aimed to make sure that PC-DOS remained one jump ahead of its better-known competitor in the version number race. When MS-DOS 6.0 was released, IBM quickly updated their PC-DOS 6.0 to PC-DOS 6.1. Soon after, MS-DOS 6.0 ran into both stability and legal problems and had to be bug-fixed several times, becoming MS-DOS 6.2, 6.21 and 6.22. In reply, IBM updated to PC-DOS 6.3, which was to become the best-known and most successful version. A substantial number of smaller PC manufacturers switched to PC-DOS at this time, particularly those that had tired of waiting for the long-promised update to the now-elderly DR-DOS 6.0 from Digital Research/Novell.
The final iteration of the DOS wars came with the more-or-less simultaneous release of PC-DOS 7.0 and Novell DOS 7.0. The general expectation was that Novell's feature-rich product would prove superior and more successful: the reality was that PC-DOS was substantially more reliable and easier to configure than either of its competitors, and usually cheaper too. In the short-term, PC-DOS looked like a winner. Microsoft, however, had not suddenly dropped out of the DOS feature war without reason: their plan was to release a version of Windows that would only run on top of their own DOS 7.0 and make it impractical to sell a competing operating system, and Windows 95 did exactly that.
PC-DOS still exists, eking out a tiny market share in the embedded market and in various other niches, and has been updated to version 8.0. This version is Year 2000 ready and is also known as PC DOS 2000. As a major part of the PC market, however, it is long past.
See also: List of DOS commands