The Quest Bed allows a Sburb player who has reached the top of their Echeladder to continue leveling up by advancing through the God Tiers. Each player has two Quest Beds, one on their Planet and one on the Moon of Derse or Prospit(depending on which planet they dream on). The trolls' session had rather than beds, though the idea behind it remains the same.
Planet Quest Beds
John, Jade and Dave's beds have been seen so far, but Rose presumably has a similar bed on her planet. For humans these Quest Beds are bed shaped stone structures marked with the symbol of the player's Aspect, adorned with a small pillar on each corner. The Troll's Quest Cocoons appear as large stone cocoons with a circular slab on the inside, which displays the player's aspect symbol. The aspect symbol can also be seen in a glowing phenomenon the locals call "The Glow Shape" hovering over the bed/cocoon.
If a player dies on their Quest Bed, or is placed on their Quest Bed after death, they are revived as a powered up self, that contains the consciousness of both the Dream Self and the regular self. Vriska seemed to have the impression she had to physically die in her Quest Cocoon to attain the God Tiers. This idea has been proven wrong by Jade being placed on her Quest Bed by Bec Noir and still gaining hers. However, a dead player has to be placed on their quest bed before their dream self dies due to reflecting the real selves injuries, so Vriska may have simply be concerned about making it to her Quest Cocoon before her Dream Self dies. This process was seen in both Vriska's and John's case, as their Dream Selves started bleeding shortly before reaching God Tier status. The resurrected God Tier self is free from any of these injuries.
The Dreamselves themselves apparently are teleported to their Quest Bed on Skaia when the real self dies on the one on their planet. John's Dream body disappears as a whole when he is awake, and is shown on the Skaian Quest Bed when his real self dies. Vriska on the other hand was awake on Prospit the entire time, and when her real self died from blood loss, her Dreamself faded from Prospit, apparently to appear on her Skaian Quest Cocoon. Jade merged with Jadesprite when she attained it, as her dream self was a part of Jadesprite. Jadesprite was, however, not teleported to a Skaian Quest bed, presumably since she was her own sentient being at the time.
Prospit/Derse Quest Beds
Aradia reached the God Tiers after the Trolls completed their session of Sburb. Aradia's dream self was in the core of Derse the entire time, sleeping in a crypt that served as her Quest Cocoon. She was a special case since she died before the session started. After her dream self was killed by Jack Noir, Aradiabot exploded, and her dream self revived.
While being placed in Derse's core was a special case, the existence of a second quest bed is not. At least each Derse dreamer (presumably Prospit dreamers as well) has a second quest bed in the core of Derse's moon. Compared to Planet Quest Beds these are mere slabs of stone, once again adorned with the aspect symbol of the player, hence they are also often called simply "Quest Slabs". They are rectangular for humans and round for trolls, mirroring the resting surface of the Planet Quest beds/cocoons. Rose and Dave discovered these Quest Beds, and were able to ascend by dying on those despite having only one self still being alive. This method of ascension does not result in a teleportation to Skaia. Aradia emerged from the exploded Derse. Dave and Rose emerged from the Green Sun an undefined time after its creation, indicating that they started at its core.