The Prison of Elders is available after completing the Queen’s Ransom mission. You must complete Level 28 before unlocking the other difficulty levels. Level 28 PoE can be completed repeatedly for rewards while Level 32/34/35 give rewards once per week. Each boss has set rounds that come before it with modifiers for each round. Once you’ve defeated the boss you’ll have access to two chests and a Queen’s Chest that can be opened with a Treasure Key.
There are a few objectives throughout certain rounds where you’ll need to do something in addition to killing all the enemies, or die:
Dismantle Mines - This requires the Fireteam to stand under a mine. One person can do it in around 10 seconds, there is a 30 second timer and you’ll have 3 mines in total to defuse that spawn 30 seconds apart. Titans with Ward of Dawn and Hunters with invisibility can usually get these the easiest on any given map.
Destroy Mines - This is much simpler than the dismantling objective. A tiny mine will float and must take enough damage to be destroyed. Three will spawn throughout the course of the wave, if any detonate you lose.
Kill Target - There is also a VIP Target that appears during certain rounds and you’ll have to defeat a Major. Heavy Weapons and Supers are best saved for dispatching these guys before they make it to all the checkpoints.