The core is essentially a "supercomputer" although that is comparative to calling the human brain a calculator. In reality the core is part computer, part consciousness, part time machine, with a little something extra thrown in for flavor. Suffice it to say the core is the most powerful and intelligent entity in the continuum. In addition to controlling the billions of biomechanical systems that allow the city to function on a day-to-day basis, the core is also actively engaged in various "thought" experiments that involve creating new aspects of reality, and modifying preexisting ones.
Originally the core was just a normal computer, it was designed specifically to answer abstract questions and solve complex problems, due to some dramatic changes in human civilization it was "upgraded" with highly speculative self-improvement technology. This resulted in countless consecutive upgrades, eventually it became so advanced that it exceeded human comprehension. Refer to this timeline
for more details.
As for location and appearance: the core is located at the heart of the diamond fortress
. It looks like a cross between a giant crystal ball and a cloud chamber.
Some trivia: in the first draft of neotoy the core was called "simuli", which was a bastardization of the word "simulation". As the connotation suggests neotoy was initially designed to be a simulation that ran in parallel to ordinary reality, not as a replacement