Generally, one runs a server with no actual graphical display attached to it (maybe a very simple one for diagnostic work). Clients connect via a network protocol, either X tunneled over SSH or a remote-desktop protocol like VNC or RDP.
With the former, users execute GUI programs from the remote shell and they show up seamlessly as windows on their client systems. This works well on high-speed networks as long as the graphics aren't intensive, but unfortunately the X protocol is very chatty and not highly efficient. It also requires each client to run an X server, which is automatic on Linux clients, easy on Mac OS, and somewhat cumbersome on Windows.
The other approach is to use VNC or RDP, which run an entire remote desktop session displayed as a window on the client. The actual work is done on the server and a compressed graphics stream delivered to the client program. There's also an in-between option called NX, which uses an optimized version of the X protocol to deliver a similar experience (with some performance improvements over VNC or RDP.) For these approaches, client programs are available for any major (and many minor) operating systems.
There is another entire way to go, though, which matches more what you are imaging: a ginormous octopus-like system extending direct graphical connections from a central server around a small area (or even a whole building). This is known as "Multiseat X", and you can read more about doing that in this article from x.org. The links from there indicate that there's enough interest in doing this to keep the idea alive, although I've never actually seen anyone doing it in my direct experience.