Windows Explorer is the file manager that comes with Windows and is not to be confused with Internet Explorer, which is Windows' default Web browser.
It is also responsible for much of the default Windows shell, including the taskbar, the desktop icons, and the Start menu/screen.
It was introduced as the default shell in Windows 95, though the old shell, Program Manager, remained as an option until Windows XP Service Pack 2, when it was stubbed out. It was completely removed in Windows Vista.
While it is not essential to the operation of Windows (one can run applications using the "New Task..." button in the Task Manager), it does make using and managing a Windows-based PC much easier.
In Windows 8, Windows Explorer was renamed to File Explorer, though its purpose remained the same. (The tag file-explorer will be remapped to this tag.)