What is a cortical circuit?
The term "cortical circuit" refers to the generalization that the neocortex is a uniform structure. For the most part, the outer sheet of the brain (the neocortex) is the same structure of neurons all the way around the brain.  It consists of the canonical six layers and it generally looks something like this.  Different research will emphasize different aspects of the circuit.

Anyway, it's quite an interesting evolutionary step for mammals.  The neocortex.. or "associative cortex" as it's sometimes called... is a very dexterous structure.  Someone born without sight will still make use of their occipital lobe to process nonvisual sensations.

Of course, as a I qualified above, it's not strictly true that the neocortex is uniform all the way around; there are exceptions.