What part of the brain is activated whenever a child enacts a particular behavior, but also whenever that child simply observes others around him/her who are carrying out the same behavior?

Mirror neurons are activated.

A mirror neuron is a special type of neuron involved when an animal performs an action which and when the animal sees other animals performing the same action. The mirror neuron replicates the behavior of the other animals as if the animal itself was acting. These types of neurons are found primarily in primates, and in humans, the region of the brain where they are located is the premotor cortex, among a few others.