What is DOS?
DOS (an acronym that stands for Disk Operating System) is a family of operating systems for IBM-compatible personal computers, very popular in the eighties and most of the nineties. And by "very popular" I mean that every x86 PC had one installed. One of the most used DOS systems was MS-DOS where MS stands obviously for MicroSoft.
With the released of MicroSoft Windows XP, software designed to run on DOS systems started to became incompatible with Windows operating systems. To properly run an old DOS game on modern computers became problematic and compatibility issues started to be the norm.
The lack of supporto for DOS based games is the reason that someone decided to create DOSbox, a DOS emulator that is still available for free. It's not the only DOS emulator, but it's by far the most popular.
For the history of DOS and more technical details, read the Wikipedia MS-DOS page.