The Silent Cave is a new Essence themed Unique Map introduced in patch 0.2 that can produce Greater Essences and empowered Unique bosses.
PoE 2: The Silent Cave Unique Map
Overview
Running the Map
The Silent Cave is an especially rare find on the Atlas. Anecdotally players have found 10-20 of the more common unique maps like Untainted Paradise before seeing The Silent Cave, and none that I've spoken to were close to the Atlas center, though that is purely anecdotal.
Like all other unique maps, completing this one will grant 2 Atlas Passive Points if you haven't yet fully unlocked the 10 passive points available for completing Unique maps.
You'll find The Silent Cave in a Desert Biome almost exclusively. The only known exception to this is if the unique map spawns close enough to a Citadel to inherit their local biomes. For example, Unique maps close to The Iron Citadel can spawn as Ezomyte City biomes.

Choosing a Map
Pro-tip: Use a Normal or Magic Waystone for The Silent Cave. There is no benefit from running rarity, pack size, quantity, or other modifiers that you'd otherwise look for on a Waystone, however you WILL be subjected to the downsides of any Waystone you put in the device.
To open the map you will need a Waystone of Tier 11 or higher. Because of the extremely chaotic nature of this fight, it's not recommended to do until you are very comfortably able to kill map bosses at the tier of Waystone you put in the atlas device.
Layout
Upon entering you'll find four entombed essences of random types of the basic variety. Each of the monsters has only one essence and is uncorrupted and they'll be evenly distributed around the center of the map.

Each of the essence monsters defeated on the perimeter will arrive on the Unique boss as a Greater essence, which are both more valuable and more useful as crafting materials. You'll always find the Unique boss entombed in the center of the map.

Vaal Orbs
These essences behave similarly to normal essences in that they can be unpredictably corrupted by Vaal Orbs, with a chance to add a new essence. Be ready though as one outcome is instantly releasing the creature inside. As you open then destroy these essences, they will float to the center of the map and empower the map boss in the center of the cave.
The downside of using a Vaal Orb is that you can re-roll the essence on the monster, potentially losing a valuable Greater. Only use them on Essences you are comfortable rolling over.
The upside is that you can double the essences and roll a second one, which will translate to a 5th or 6th Greater essence on the unique boss.

Pro-tip: Most of the time it's a good idea to use a Vaal Orb on these Essence monsters unless there is a particularly valuable Essence on the monster.
Check the auction house before committing as prices vary from league to league, but some consistently useful and valuable Greater essences are
- Greater Essence of Haste
- Greater Essence of Ruin
- Greater Essence of Battle
- Greater Essence of Torment
Unique Boss Fight
The Unique map boss empowered by the the Essences can be extraordinarily strong because of the overlapping and chaotic nature of Greater Essence modifiers. Make SURE you are comfortable fighting map bosses before attempting this map.
To demonstrate exactly how chaotic a map boss with several essences on it can be, here's a section of a fight with Zekoa, The Headcrusher with five essence modifiers.
Pro-tip: Use Magic or White Waystones as the loot you'll drop is wholly independent of rarity, pack size, quantity, or other factors that would normally play into choosing a Waystone.
Speculation
Anecdotally, it's believed Atlas Boss Passives play a role in how difficult the Unique map boss is in The Silent Cave. I've not yet been able to confirm this empirically but will update this guide as soon as I have concrete information.