Linux desktop environments are the graphical interfaces used to interact with the operating system.
It's important to note the difference between desktop environments and window managers. Window managers only take care of application windows themselves - how they look, what buttons are shown (minimize, close), how they are arranged, etc.
A desktop environment on the other hand includes more of a complete suite of managing a desktop. For example, a system tray, system menus, backgrounds, icons, and a task bar. They often include applications as well.
Desktop environments usually come with a window manager of their own, but the window manager of any desktop environment can be substituted for another.
A good Linux desktop environment enable interacting with the operating system easily and efficiently. It should offer a intuitive, efficient interface for getting things done that can be easily customized to fit everyone's needs.