Oh, you mean the vomiting-up-souls thing? It comes up in chapter 48. A dragon can put his/her mind and soul in a sort of gemstone that exists inside the dragon's body. Then the dragon can 'expel' the stone from his/her body, in which case killing the body will not destroy the soul. Galbatorix's unnatural power comes from these 'heart of hearts' that he's stolen and bent to his will.

Glaedr ends up giving Eragon and Saphira his heart of hearts, so we get to 'see' the process. Unfortunately for the drama of the idea, the actual 'expelling' resembles nothing more than a cat throwing up a hairball. (Or at least, that was my first appalled thought.)