By specifying the DISPLAY variable you can launch the application at any valid endpoint, e.g. DISPLAY=serverhostname:0 xterm
This command launches the program xterm on your local machine (client) but forwards the display to the server.
When you take a closer look at your usual setup where server and client on the same machine you notice that usually the environment variable DISPLAY is set to ":0" which just means it is display 0 on your local machine.
Edit: It is actually very useful for remote administration of tools that need a gui or offloading the program on another machine. Unfortunately it isn't used that much anymore and wayland ditches the support altogether and instead the compositor or some other layer has to provide this feature. Note this is very distinct from vnc, rdp etc, because it works per application and fully transparent from a users perspective.
]]>You do not need an X11 server on the machine you're running your desktop environment (or any X11 client) on.
Yes, it's a common setup, but not required.