What is the idea or belief that people have a fixed nature that makes them who they are and fully defines them?

The general philosophical term for this is "determinism." Philosophers have, since the very origin of the discipline in classical Greece, debated the question of the degree to which we are determined by external circumstances or some fixed aspects of our natures versus the degree to which we are autonomous and have free will. Religious thinkers are also concerned with the problem of how humans can have free will if God or gods are omnipotent and omniscient and predestine us to salvation or damnation. 

The question of free will versus determinism is especially important when we think about ethics. For example, if someone is mentally ill and commits a crime, we might argue that the person is not in control of his or her actions and thus is not legally culpable. As biology, psychology, and medicine make new discoveries about the nature of our brains and behaviors, increasingly it can be argued that psychopaths or other types of criminal may be hardwired to act in certain anti-social ways, while other people may equally be predisposed to what we view as morally good forms of behavior. If our actions are determined by genetic and environmental factors rather than free will, we need to rethink our ethical systems to somehow take these issues into account.