Tartaros is for the extremely bad. Persons such as Kronos are placed in Tartarus. It is meant to hold the most dangerous and bad of all who end in the Underworld. Most monsters are placed there once they "die" until they re-form.
Fields of PunishmentEdit
The Fields of Punishment is for those who are bad once in the Underworld. Punishments vary depending on how bad of an act the person or persons commited.
Fields of AsphodelEdit
The feilds of Asphodel is where those who are normal (in other words don't do anything remarkably good or bad) go in the Underworld after they die.
Elysium is where you go if you've been judged and deemed good. You are also given a choice (if you've been deemed good) whether you go to Elysium or try for rebirth. If you go for rebirth, live again, get deemed good in all those lives, you got straight to the Isles of the Blest, which is the best place to go in the Underworld. It is said by Percy that it is like a gated community, with golden gates, and laughter inside. He also states that you can smell barbecue cooking from inside.
Souls can only go to Elysium when they go to the council. Most souls don't want to take that risk and just go to the Fields of Asphodel.