Compiz is a composite window manager with very little desktop capabilities (no iconbar, application menus, etc). It is designed to be used as a drop-in replacement for other window managers in today's desktop environments, such as KDE and GNOME, which provide these desktop utilities.
As a composite window manager, Compiz allows the mangling of the windows themselves. This is accomplished by use of 3D, and as such, Compiz requires 3D acceleration, as well as a supported X server (either one that uses AIGLX or the XGL server). Even the most basic of 3D acceleration is likely to be enough to run Compiz in a acceptable manner, although some visual effects may not be usable. All the effects that Compiz can perform are implemented as plug-ins, and the window decorations are handled by a 'window decorator' rather than Compiz itself. All the plug-ins can be activated or de-activated separately, and have actions bound to various key-combinations (which most plug-ins use to activate or de-activate their effects, at the very least). All key-combinations are customizable, along with various other preferences.
Compiz Fusion is a community project spawned out of Compiz's history. It contains its own window decorator, called Emerald, and several plug-ins. Both projects are currently (as of 2008) being developed, with Compiz handling the core functionality, and Compiz Fusion being more user-orientated and developing add-on plug-ins, themes, etc, in a effort to make things more user-friendly.