Any DE (Desktop Environment), and most WM's (Window Managers), in Linux have multiple desktops. They've just never been able to "cube" it like that before.
Compiz is a WM replacement. It replaces whatever your DE's WM is (Gnome = Metacity | KDE = KWin, XFCE = XFWM4, etc). Instead of having your normal WM controls the windows, you are handing it over to Compiz.
Compiz has its own "themer" called Emerald. This is what decorates your windows (the styles/themes). Emerald themer only themes windows built by Compiz, so you can't use it with your normal WM.
Compiz Fusion is a set of plugins that add extra controls that only the Compiz WM can handle, like cube, scale, rotate, rain, fire, wobble, etc etc. There are a good number of plugins, all doing separate things. However, some plugins rely on other plugins to be loaded (like rotate, it obviously needs a 'cube' to rotate, etc). when everything is working together it is a perfect mesh of never before seen eye-candy that is hard to ignore, even by Windows users.
Multiple monitors is not part of the DE or WM, thus Compiz doesn't know or care if you have 1 or 16 monitors. It just sees a DE that needs a WM. The multiple monitor setup is all in the Xorg backend. However, when there are multiple monitors there are all sorts of things you have to take into account, like hardware texture limits (ATI's is 2048x2048 - nVidia's is 4096x4096). This limit just means anything bigger than this size will not be rendered on the screen. Things will get cut off. Like my desktop wall paper. If I set my desktop resolution to 2056x1024, then on the far right side I get a huge white space - the wallpaper just stops drawing where the hardware texture limit is. However, if I am not running Compiz, the background shows up fine - all the way to the edge. Why?
Compiz is not just another WM. It is a WM that doesnt use the FB (frame buffer) or hardware to draw to the screen - in fact, it uses OpenGL. Everything being drawn gets fed to OpenGL and then OpenGL places it on the screen - this is the reason you can still have full DRI while using some of these effects (like watching a movie while the cube is rotating, without ever missing a single frame). Since OpenGL is the main tool here, the texture limit comes into play. The hardware texture limit is the maximum size any 3D object can be before it just stop rendering.
That's just one of the things to take into account when using Compiz. Its actually a very picky task. Some people find it extremely easy to install and use, while others have the hardest time. It just depends on your situation/setup/hardware/whatever.
For more info you can check out their project site at
http://compiz.org/ - you'll find a lot of info there, including screenshots & videos, and a link to their forums.
