How does the human brain store memories?

Memory in the brain arises from changes in neural circuitry. When we learn, neurons form or strengthen synaptic connections (via long-term potentiation or depression). The hippocampus helps encode new memories and then redistributes them to the cortex for long-term storage. In essence, a memory corresponds to a pattern of strengthened (or weakened) synapses across neural networks. Those networks of connected neurons represent information; whenever you recall something, that circuit is reactivated. Thus the brain "stores" memories by altering the strength and patterns of millions of synaptic connections.