How Memories are Physically Stored in our Brain
We know that the hippocampus in the limbic system in involved in the process of memories. For example, damage to the hippocampus can cause anterograde amnesia. People with anterograde amnesia cannot encode new memories but remember ones already in their long term memory. Sometimes they can learn new skills, but will never remember having learned them, showing us that procedural memories are not handled by the hippocampus. But memories are actually stored throughout our whole brain.
The current theory on how memories are stored on a neural level is called long-term potentiation. Studies show that neurons can strengthen connections between each other. Through repeated firings, the connection is strengthened and the receiving neuron becomes more sensitive to the messages from the sending neuron. In other words, it is kind of like using a machete to clear a pathway (in the case neural) in a jungle. The first time you chop through (fire) it is a slow and messy process. Like trying to memorize a song for the first time. But the more times you chop through that path, it becomes clearer. Eventually, you have a clear path (strong neural connection) and a great memory- it is rehearsal for our neurons!