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.
Upstream Compiz Maintainers