Windows is a proprietary operating system which you cannot compile or modify at a source code level. This allows Microsoft to maintain control of their operating system and many people disagree with the notion of using closed source software.
ReactOS is a free and open-source operating system, which has been in the alpha stage of development for about 10 years, which allows Windows applications and libraries to be executed without having to use a single piece of Microsoft-owned code!
A similar project to ReactOS is WINE, which allows you to execute Windows binaries in Linux and OS X.
ReactOS has an easy-to-use graphical interface, which resembles Windows Classic. Some things, like the Windows logo, are avoided however. It adds some features Windows lacks by default, such as virtual desktops.
The practical uses of ReactOS are somewhat limited. The operating system can run many Win32 applications, but this is patchy, and support for .NET Framework applications, OpenGL and DirectX is still being developed. Their own website acknowledges that ReactOS is 'not recommended for everyday use'.
Their website boasts a selection of screenshots of all sort of programs running under ReactOS, for example OpenOffice.org, Mozilla Firefox, VB5, Winamp, Diablo 2 and Unreal Tournament!
The operating system is probably not usable as of yet as someone's main operating system, but its idea and sheer curiosity value makes it fascinating. When it becomes stable, it will be interesting to see how a completely Win32-compatible operating system fares against Windows XP and Windows Vista.
The project maintains a compatibility database that you can use to find out if a program that you are using is supported by the operating system yet. It currently lists less than 1000 applications and drivers which may not sound like a lot but features prominent programs such as Firefox, Cpu-Z or Nvidia display drivers.