Rebirth

"You do not understand the role of Chaos. The Chaos that the Deity is made up of can drown the world, but it is also the Key to our Humanity. When all things die, we return to it...but only for a short time. When a Man dies, his soul melts into the Chaos, but the idea of it-of him-survives intact. And then, he is reborn anew, and returns to the living world. An unseen power exists in all of us. It is the Chaos, that we carry in our hearts."

- Zagreus explaining the nature of the Underworld, of inevitable Human Rebirth and the nature of Humanity, and how it connects to the Protogenos Chaos in the Blood of Olympus.

Rebirth is an act in the Cycle of Death and Life, despite what is implied it is actually revealed that all of Humanity are inevitably forced to go through this cycle, only those in the Isles of the Blest are granted Immortality and amnesty.

Characteristics
The Underworld is a location encased in the Chaos, enveloped by the stretching arms of the Protogenos Chaos himself, the souls of all living things including Chaos's own children are forged from fragments of his Dark matter that carry the same name. When a human dies, their soul melts back into Chaos's welcome arms but the idea of him survives giving birth to Underworld Spirits and Ghosts, these entities gradually lose their memories and knowledge of life until they are deemed sufficient enough to be reborn from scratch.

Each circumstance of Rebirth is different, those with stronger willpower or are unable to let go of the things that anchor them to life take far longer to be broken down to their "factory settings", e.g. many Heroes and Demigods, including the Original Jason, Orpheus, etc.

Some Ghosts, like Achilles, are bound to the Realm of the Dead to guard a point that had been monumental in their lives such as Achilles guarding the River Styx and passing on a warning. Those sent to Elysium are somewhat immune to the Rebirth Cycle, however they are offered Rebirth and human nature leads to many accepting, those that reach the Isles of the Blest are granted Immortality and thus are immune to being reborn.

When those in Elysium choose to be reborn and it is granted by Zagreus, they are taken to the Cave of Hypnos where they are encouraged to drink from the Lethe and placed into a deep sleep, Hypnos and Thanatos then bathe that person in the Lethe which accelerates the erasure process of the individual's past life. From there, Zagreus turns the person as a Spirit Wisp once they are renewed back to their base form, which he takes to where Erebos and Gaea meet to be woven with Chaos by Gaea to become a soul and then taken by her to be reborn within a pregnant woman.

Immortals themselves can also be reborn, the Titans and Gigantes return to Tartarus where they are reborn and it is implied that the Gods would return there too, when a God loses all purpose they fade back into Chaos and become one with him thereby dreaming the long dream that Chaos has of the living world.

Trivia

 * Rebirth is the process in which Humanity are reincarnated.