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The core is the unconscious part of the brain, responsible for most of the operation. It has full, executive control over everything. It controls hormone levels, breathing rate (though this can be delegated), heart rate, organ activity, mood (happy? sad?)1), sensations (hunger? thirst?), level of consciousness, metabolism, pretty much everything you do not have conscious control over.
Information from your senses go to it - eyesight, hearing, haptic information, taste, etc. go to the core. The core then decides which to focus on, which are important and which can be ignored.
Conscious thought emerges from the core, we remember things not because we can consciously go through our memories, but because the core pulls that information out of storage. I think that our consciousness is a phenomenon that, ultimately, sits below the core, and any actual thinking happpens at the core. We basically just get to watch our brain think, but (because of how complex and unintuitive our brain is) it looks like we are doing the thinking ourselves.
As mentioned before, the core is responsible for all kinds of system tasks - heart rate, maintenance, immune response2), body temperature, resource management (energy and water in particular), etc. etc., but it does a lot of Direct Engine-ey things as well. One very important one of these background tasks is what I could best describe as a comb. As I live, I constantly make new experiences and learn new things. The “comb” goes back and forth through my entire memory storage and checks the new information against the old. Add to old experiences where it fits, identify inconsistencies where it doesn't and, hopefully, learn something new. This is a constant process.