Do I need X11 to install other desktop environments?

X11 and desktop environments play different roles.

X11 is a display server protocol. The most common display server implementing this protocol is X.org. It combines the following main features, from the lower level upwards:

• video and input drivers;
• a canvas to draw on (taking orders like “draw a light green rectangle at these coordinates”), and input events (generated from keyboards, mice and the like);
• a notion of window, with each window getting some portion of the screen real estate, a stacking order of windows from background to foreground, and one window at a time getting input events (which is known as having the focus).

X11 has no viable alternate (though Wayland may get there one of these decades). It's part of every Unix system's GUI layer.

On top of X11, you need a window manager. The window manager handles things like decorations on windows (borders, title bars, etc.), window manipulations (moving and resizing windows, icons, etc.), focus strategies (e.g. focusing a window when you click on it), etc. X11 contains a built-in window manager, but it's extremely crude.

Some, but not all, window managers are part of a desktop environment. A desktop environment provides additional features like ways to launch programs (e.g. menus), applets (clock, workspace switcher, configuration interface, …), and so on. Desktop environment often come with a suite of applications that have the same look and feel but can be used independently as well.

Gnome, KDE and LXDE are all desktop environments. They come with a window manager (Gnome gives you a choice, KDE and LXDE have their own).