- Enlightenment: your desktop "interface", what manages entirely everything that runs on the graphical system, alternatives: kde, gnome, xfce, etc...
- Composite: is a "layer" in the graphical system, let's say between your desktop (enlightenment) and your screen that modify's the visuals, like adding transparencies, etc... can be used accelerated by hardware or by software (using the cpu instead the gpu), which is best compatible and working in any computer, no matter which graphical card the person has
- E-module: is like plugins or extensions for your Enlightenment desktop, one of them is Composite, or Ecomorph, Engage, or some Elive own plugins too, like emodule-internet
- Ecomorph: is a fork of Compiz made to work with Enlightenment, so its a compiz very different that has been renamed.
- Compiz: compiz is like a composite layer using strictly 3d-accelerated hardware, adding nice visuals, there's not much distros that still uses is today even if looks and works good, Elive uses it in the form of Ecomorph
In short, you can turn on/off anything that you want 
Mmmh, all the listed things are pretty exclusive of Elive, so debian has not it by default, so the answer is: newer elive with newer debian base should take care to make all compatible or to implement their own alternatives
PD: this is a kind of glosary documentation, where we could publish this information easily to find for who wants it?
