X11 is the window protocol used by most Linux/Unix and Sun systems. Many popular Unix programs run, or at least originally ran, only in X11, such as OpenOffice/StarOffice, The Gimp, KDE, Gnome, etc. While a few of these have been ported to the Mac's windowing system (Aqua), most still run under the X11 service instead.
X11 has a few unique abilities. Provided you have a fast network connection, you can remotely bring up a particular program's window, kind of like using a remote desktop or VNC client, but instead of showing the whole screen it only sends the windows you are using. X11 can be tunnelled in this way through different network ports or through an SSH secure connection. It is also highly expandable, and is used to drive many video-walls and other obscure applications.
A popular open source version of X11 is XFree86, which you'll find on many Linux distributions.
That said, there's not a lot you can "do" with X11 as such, its better to think of it as being the platform on which some Unix apps run. You would probably not want to develop for X11 if you want to run on Mac, as the Aqua windowing system is far superior. You'd only look at developing for X11 if you wanted to run programs on Unix/Linux.
If I were you, I'd install Fink and FinkCommander and try out some good, popular X11 window managers like KDE, Gnome or Enlightenment.