The human brain has roughly 86 billion neurons connected by an estimated 100 trillion synapses. It is organized hierarchically: a brainstem that keeps you alive, a limbic system that handles emotion and memory, and a six-layered cortex that does most of the heavy cognitive lifting. The Society for Neuroscience and the National Institutes of Health publish freely available primers that go much deeper than this.
Neuroplasticity - the brain's ability to rewire itself in response to experience - is now established at every age. Pioneering work by Michael Merzenich (UCSF) and Eleanor Maguire (UCL, the famous London taxi-driver hippocampus studies, 2000) showed structural changes from learning that were once thought impossible in adults.
Memory is not a single thing. Working memory (Baddeley, University of York), episodic memory, semantic memory, and procedural memory each rely on distinct neural circuits - which is why a person can lose one and keep the others. Understanding these distinctions is essential for evaluating any 'brain training' claim.
Most popular brain myths - that we only use 10% of our brain, that we are 'left-brained' or 'right-brained' people, that vaccines harm brain development - are not supported by neuroscience. The Dana Foundation and Neuroscience News maintain ongoing myth-busting resources written by working researchers.