Ubuntu 11.10 by default only comes with a combination of Gnome 3 and Unity, available in both 3D and 2D variants.
If you want Gnome Shell instead, you'll have to install it. And same of course goes for any other desktop environment than Gnome.
If you want Gnome 2, then no, you can't get that any more. It's out of development and unsupported now. Gnome Shell's fallback mode is quite similar, though, as is XFCE4 and perhaps even LXDE.
Anyway, you of course don't need to install Compiz to change your desktop environment. Not that you even could install Compiz, it's already installed by default like it has been for several Ubuntu versions now. And assuming you were talking about CompizConfig Settings Manager, that's also unrelated to changing your desktop environment. They are simply selected from the login screen..