The brain cortices (plural of 'cortex') are areas of specialization within the brain, typically dedicated to the relatively higher functions of brain activity. Generally, the word 'cortex' simply indicates the outermost layer of an organ, and this is true as well for the brain cortices. Functionally speaking, there is only one brain cortex, the cerebral cortex. However, this is further subdivided into other functional (and purely definitional) brain cortices. The whole of the cerebellum is made up of a deeply folded cerebellar cortex, but when we are speaking of the brain cortices we are typically referring to the cerebral cortex.
Most of our higher functions, those human abilities that differentiate us from the other animals, are coordinated within the cerebral cortex. This is the outermost layer of the cerebrum, wholly composed of gray matter and subdivided into six sub-layers, from the hippocampus to the neocortex. Even these divisions are further divided, for instance into the motor-sensory cortices. The motor cortex is the side of the brain that controls voluntary muscles. The somatosensory cortex is the side of the brain that receives and analyzes sensory impulses from all parts of the body. The motor cortex governs the body's response to messages from the outside world, and the motor cortex in the right hemisphere controls the motor responses in the left side of the body while the left hemisphere controls the right side.
Additionally, you have the prefrontal cortex around the brain's frontal lobes. Here the executive functions of the brain are carried out, including planning and decision-making. Other primary brain cortices include the visual cortex, at the back of the brain, and the parietal cortex located forward of it, which it is critical to interpretation of sensory data.
Yet the brain cortices are being continually further subdivided with further progress in brain mapping and with our greater understanding of brain functioning.