Window manager
The program that decides where windows go, which one has focus and what the frame around them looks like.
In X11 the window manager isn't part of the server. It's an ordinary client with special privileges: it gets notified when an application wants to map a window and can move it, resize it or refuse it. Most also wrap each application window in a frame of their own with a title bar and buttons, a trick known as reparenting. The split was deliberate. The X designers refused to dictate what an interface should look like — "mechanism, not policy".
The upshot was hundreds of window managers, from twm and mwm to today's tiling crowd. Under Wayland the role is merged with the display server into a compositor, and swapping one out on a whim is largely a thing of the past.