What is Compiz ?
(taken from Compiz README)
Compiz is an OpenGL compositing manager that use GLX_EXT_texture_from_pixmap for binding redirected top-level windows to texture objects. It has a flexible plug-in system and it is designed to run well on most graphics hardware.
Okay, this description isn’t really meaningful. In a nutshell, Compiz is a compositing manager, which means that it enhances the overall user interaction by adding fancy effects to your windows, from drop shadows to awesome desktop effects like the Desktop Cube or the Expo view.
Compiz can also be a window manager, which means that it is the software between you and your desktop apps. It enables you to move or resize windows, to switch workspaces, to switch windows easily (using alt-tab or so), and so on.
There are two release series of Compiz available that differ in certain technical respects although functionally almost identical. Plugins built for one series will not work with the other.
The latest stable release of the Compiz 0.8 series is 0.8.8.
What was Compiz Fusion ?
If you heard of Beryl or Compiz Fusion, it’s us as well ! Check the wiki page for more details on the (somewhat troubled) project history.
(taken from AboutCompizFusion Wiki page)
Compiz Fusion is the result of a merge between the well-known Beryl composite window manager and Compiz Extras, a community set of improvements to the Compiz composite window manager. Compiz Fusion aims to provide an easy and fun-to-use windowed environment, allowing use of the graphics hardware to render each individual window and the entire screen, to provide some impressive effects, speed and usefulness. The first Compiz Fusion developer release was Compiz Fusion 0.5.2 on August 13th 2007, shortly after Compiz 0.5.2 was released. First stable release of Compiz Fusion is 0.6.0 released on October 20 2007.
Compiz is an open-source software project, meaning anyone can use it freely and contribute.