All sentient beings have to have something that gives them life. Humans have their souls, angels have their halos (actually, there is only one halo, which they all share. reminde me later and i might write more about the halo? its really kind of confusing). Demons are unique from angels and even fallen angels because their life comes from their aedifex.
The aedifex was created by the blood magic of The Satan, Lucifer, after his fall from grace. At first he made other demons 'from scratch' as it were, flesh costructs given life from the aedifex. These were Paimon, and Azazel. He used these first creations as experiments, because the true purpose of the aedifex was to cheat death. You see, dear reader, even immortals must one day die. In the grand scheme of it all they are too young to know the limits of their lifespan, whereas humans know it very well. But even the most perfect angel, if she does not fall in battle, will grow old, and will die. And given The Satan's nature, he has many enemies. If he were to die, his kingdom would slip from his grasp and crumble into nothingness. But if someone could bring him back?
You see where I'm going with this, right?
So if you take all of this into account, it becomes very plausible that the aedifex was created with the specific intention to be able to preserve someone's life after they died. They are able to enter bodies very easily (which is how humans can become demons with relatively little fuss) and they are said to retain the traits of their original owner for some time after death.
So an aedifex is what makes a demon a demon. It affects their bodies in strange ways, gives them powers and strength beyond our comprehension, and the longer they have it, the more familiar with it they become. A very old demon can manipulate the power stored in its aedifex far more than a new demon. So how does this allow a demon to cheat death? Well its pretty complicated.
After a demon dies, it's aedifex leaves its body. Depending on the strength and age of the aedifex, it may either die or return to the collective aedifex, which is basically a big cloud of the things tended to by an opifex. The only known opifex is Amrondre Castor Nouo, so his name (castor) has become interchangeable with opifex. In the collective, the aedifex stays there until it has lost enough of its memories and regained enough strength to be reused in a new demon.
For the process to work, the body's living tissues must be very well preserved. If not, a healer must be present to rebuild the tissues and make them habitable. This is, as far as I can tell, impossible for most healers. The texts claim that a very very talented healer can rebuild an entire body from nothing but a lock of hair-- and given that The Satan was able to create entire bodies with consciousnesses and everything from nothing at all, i presonally choose to believe that's true.
The Artifex's job is to summon the aedifex and guide it into the body. So if you have the body of the deceased properly prepared, a sufficiently talented artifex, working in tandem with the opifex/castor, may be able to summon the *exact* aedifex that just left the body, and guide it back into the body. Essentially resurrecting them. There seems to be a very wide margin of error in this process, especially since it seems to require at least two very skilled people working in tandem, very soon after death and under very specific circumstances.