The neocortex is the newest part of the cerebral cortex to evolve (hence the prefix "neo"); the other parts of the cerebral cortex are the paleocortex and archicortex, collectively known as the allocortex. The cellular organization of the allocortex is different from the six-layer structure mentioned above. In humans, 90% of the cerebral cortex is neocortex.
The cerebral cortex and the neo cortex are the same thing. The cerebral cortex is the outer layer of the brain. It consists of 6 cell layers. It was that last part of the brain to evolve, hence the term 'neocortex'. The cortex covers the whole brain, not one side or the other. It is the crinkley surface that you see in pictures of the brain (many convolutions; gyri and sulci).
The cortex is associated with 'higher' functions; consciousness, language understanding and production, cognition, abstraction, problem solving, motor and sensory integration and so-on. Things like love and empathy are more likely to be associated with the limbic regions, which are considered the 'emotional core' of the brain.