Cerebral cortex is just the outer shell of the brain, a rind about a tenth of an inch thick. It is deeply infolded, however; if you peeled it off and flattened it out, it would cover about four sheets of paper (in humans; for apes, only one page; for monkeys, a postcard; for rats, a postage stamp). By weight, it's a very small part of the brain, but it is in charge of linking up new things with one another. You often need it to learn something new, but deeper ("subcortical") parts of the brain may suffice to do it for the tenth time.
In particular, cortex is very important for all the things that we humans can do better than the apes. The higher intellectual functions are mostly examples of structured thinking, not just simple association.
We know most about the structuring from Chomskian linguistics, because it tells us a lot of structural things that are impossible for the brain to do in any language.
Nothing like higher intellectual function is seen in the great apes. Perhaps, with sufficient tutoring in early childhood, they will turn out to have hidden abilities (just as humans, with tutoring, can learn to read, even though it isn't "natural" in the way that acquiring words and syntax seem to be).
The parts of the cortex that do all this structured stuff are what's beneath your left hand if you place it on your left forehead, with the thumb behind your left ear and your fingers stretching over to the right forehead (well, assume you have long fingernails).
And remember that cortex is in charge of novelty, not routine. It's got to plan sentences and agendas that are coherent, starting with a jumble of things that don't fit together very well (just like your nighttime dreams) and shaping them up into something that makes sense – and wins a competition with other thoughts still cooking on the back burner when it's good enough, then emerges from your mouth. This quality improvement process may work a lot like the one that Charles Darwin discovered in 1838 -- except that it doesn't take millennia. The brain has the right circuitry (see my book, The Cerebral Code) for performing a Darwinian copying competition where your memories are the "environment" that judges how well new things hang together.
William H. Calvin, Ph.D., is a theoretical neurophysiologist at the University of Washington in Seattle (http://faculty.washington.edu/wcalvin) and the author of such books as How Brains Think and The Cerebral Code.