The Monkey Brain and the Computer are simple metaphors for the two thinking systems described by Daniel Kahneman as System 1 and System 2.
It is first mentioned in The 7 Skills to impress™: Part 1, Chapter 3.
The Monkey Brain corresponds to System 1. It is fast, automatic and emotional. It scans constantly for danger and reacts instinctively to protect us. Under pressure, it drives fight, flight or freeze.
The Computer corresponds to System 2. It is slower, deliberate and logical. It allows us to reason, plan, weigh options and make considered decisions. When we are calm, the Computer stays online. When stress rises, the Monkey Brain can take over and suppress it. The 7 Skills are designed to calm the Monkey Brain so the Computer can function effectively, especially under pressure.
See also System 1, System 2.