Aradia finds a dream bubble
Dream bubbles are found in the Furthest Ring, glubbed into being by Horrorterrors. The Horrorterrors were convinced by Feferi to set up special bubbles for our protagonists to use, including the pre-scratch trolls. She was capable of negotiating with them due to her previous experiences with her Lusus, G'lbgolyb (Emissary to the Horrorterrors and possibly one herself), having gotten her used to the nature of the Horrorterrors, as well as a possible method of directly contacting them (as Feferi's dream self was cut in half and her original body later had a hole blown through the chest, she is probably rather glad that she set the system up).
Purpose
Jake meeting Aranea in a dream bubble.
The bubbles allow the players of Sburb/Sgrub who have lost a life to continue dreaming. The reality constructed in the bubbles is a combination of the memories of everyone inside the bubble accessible to any who has learned the bubble's nature. According to Aradia, it would be quite difficult to exit a bubble to get into a new one, however someone inside a bubble can travel to another one through common points in memory, taking their own memories with them to be integrated into the new bubble. These memories mix and combine, but in general no new memories are created.
Various dead trolls about to meet their death at the hands of Lord English.
Additionally, the bubbles store the consciousness of a player who has lost their dream selves and their original selves, making them double as a form of afterlife; the player is then capable of accessing the memory of anyone else who has gained access to the bubbles. Living players, who have lost or become their dream selves, appear in the dream bubbles when they are asleep. It is in this fashion that the bubbles can be used as meeting places between the dead and the living; even those from alternate universes and doomed timelines can be found in a bubble. Furthermore, in addition to memories, the bubbles can to some extent reflect the thoughts of a player. Beyond that, the bubbles can also behave similarly to Skaia's clouds, showing things that are in some way relevant to the player - such as when coin-flip Dave saw alpha Dave grieving over Bro's corpse - something he was not aware of before being shown it by a bubble.
As demonstrated by Aradia, a player who has died prior to entering The Medium will not have their soul taken into a dream bubble, an event that also renders their dream self near-inaccessible as they will be located in the core of their associated kingdom's moon and will seemingly not wake up during normal gameplay.
A final purpose
posited by Aradia and Vriska is possibly the entire point of the existence of the bubbles: they were created by the Horrorterrors as bait for Lord English, to temporarily get him off his self-assigned task of destroying them. If this is true, defeating Lord English would make the dream bubble afterlife pointless to maintain further.
Mechanics
Rose and Dave in a dream bubble.
The bubbles can be physically entered as well, as Aradia was capable of pushing herself through a bubble's skin to observe the battle with Bec Noir. Time and space are unstable in the Furthest Ring and mastery of at least one is required to navigate it, making it difficult to actively seek out dream bubbles. Roxy has been shown to enter a bubble with ease, subconsciously navigating the Furthest Ring due to her Hero of Void powers.
Rose was able to accomplish a similar feat by piloting the severed moon of Derse into a bubble, although the program used to navigate was created by Dave, a hero of Time. The trolls' ectobiology lab meteor has also passed through multiple dream bubbles that happened to be on their course through the Furthest Ring.
Dirk trying to exit a dream bubble.
While it is possible to physically enter and exit dream bubbles, the actual inhabitants of a bubble (i.e. dead and dreaming minds) are usually not capable of leaving it. This was demonstrated when Dirk's dreaming consciousness was unable to leave a bubble while the physically present Roxy and Serenity could. Aradia also remarked on this when Sollux was actually able to leave his bubble, a feat she previously considered impossible. It is unknown if memory objects can leave the bubbles, However they likely can, as the Cod Piece was from Gamzee's outfit was found by Found by Meulin in a Dream bubble, although it just as likely could have been a physical object.
Apparently, items found in dream bubbles can be taken back to the physical plane. Kurloz gave Gamzee a codpiece in a dream bubble and he was later (from Gamzee's chronology) seen wearing it on LOCAH. Additionally, it appears that dead characters present in the dream bubbles can have lasting effects on the characters who are physically passing through the bubbles, such as when Feferi healed WV.
Aranea was also able to leave a dream bubble with Gamzee after being resurrected by the Ring of Life, implying that players dreaming in a bubble may be capable of taking anything they are in physical contact with back to the waking world with them; however, the rules governing this are unknown – it may be limited to certain objects (particularly the Ring of Life), to god tier players, or other unseen factors.
Alternate timeline selves perishing at the hands of English.
Dream bubbles can be destroyed, as demonstrated by Lord English. He also was able to permanently kill all of its inhabitants. It is unclear if the former is required for the latter and also if Lord English is the only one able to perform these feats, though several characters speculate that it probably is a power exclusive to him.
It seems that dream bubbles can be put to more unusual purposes. Smaller, miniature bubbles can be used to store memories. Dave has used these as a replacement for the Internet, creating a parody of a Twitter spambot that he calls dave_ebubbles. It regurgitates everything he's said over the course of the adventure. There is a real equivalent on Twitter, which can be visited here. Similar bubbles are also used as switches that alter the surrounding area by releasing the memory inside. They can only be used by the owner of said memory.
The Horrorterrors have set up a method of communication for the inhabitants of each dream bubble: a social network called Bubblr where players can "rebubble" memories to a potential audience of everyone in the afterlife, edit their memories before they send it to Bubblr, follow other players, or leave comments on memories. The memories that players can send out apparently stem from their souls, and Bubblr has many other ties to each player's soul.
According to [S] ACT 6 INTERMISSION 3, the Horrorterrors act as moderators for Bubblr. Their duties include suspending players they view as breaking the rules of Bubblr from using it. If everyone in the afterlife follows any given player through Bubblr, that player will come back to life. Conversely, if a player ever has zero followers, their soul dies forever.
Entities called spamghosts exist in the dream bubbles that can hack into souls and send out bizarre advertisements to other players.
The bubbles, despite being a relatively recent creation, have technically existed since at least the time of the Pre-Scratch Troll session, given that Meenah knew about them. This is due to the Furthest Ring's bizarre temporal properties.
Notable Bubbles
The planet found inside the dream bubble visited by Roxy.
The very first dream bubble seen was included in [S] Jade: Wake up. It was very unstable and, at points, nightmarish. Later on, Feferi would ask the Horrorterrors to set up a series of stable dream bubbles.
In one bubble (colored blue) an entire planet was seen, populated with alternate-timeline dead trolls and visited by god tier Rose and Dave, as well as a sleepwalking Dream Roxy. The planet itself has features of Earth, Alternia, Skaia, all four planets of the B1 kids, and some of the A2 trolls' planets (LOQAM, LOPAH, LOLCAT, LOTAF, and LOCAS). This bubble and apparently all its inhabitants were destroyed by Lord English. (The meteor travelers' memories made up some of the landscape, but they escaped the destruction of the bubble.)
Another bubble (this one pink) features extensively starting in Act 6 Act 2. This bubble is where the Beforan trolls now reside after their demise. It was first discovered when Jake blacked out and saw Aranea, and again when Roxy got trapped between two Fenestrated Planes and was attacked by Meenah.
Post- and pre-scratch trolls and kids meeting for the first time in a dream bubble resembling Alternia's pink moon.
The dream bubble seemed to largely feature the pink moon of Alternia, but was later shown to have aspects of other areas as well, including Prospit, Derse, areas that resembled LOMAT (likely Aranea's planet), LOBAF (possibly Mituna's planet), and more (including Karkat's neigborhood and Terezi's forest). It is where the Act 6 Intermission 3 walkarounds take place.
A third, colored pale blue, was seen when Jack Noir was searching for a place to hide from PM and rest. He entered this bubble, which took on his memories of the death and destruction he caused in the A2 and B1 sessions. John and Rose gained access to this bubble through their shared memories with Jack, it is later revealed to be the bubble inhabited by alpha-timeline Tavros and Vriska which is later revealed to be the same one as the one Openbound takes place, or to have merged or be intersecting with it, or for Vriska, Tavros and John to have switched from their bubble to the Openbound bubble.
Alternate Calliope was later found by Calliope and Jade: She had been holed up in what appears to be a hidden and isolated dream bubble for “a very, very long time” as a result of her deal with Echidna, waiting for Calliope to arrive and serve as her signal of freedom.
Notes/Speculation
An external view of a dream bubble.
Several dream bubbles seen so far have had a planet at their center. This may be true for all dream bubbles, but that is unknown as only a few have had their interiors shown zoomed out enough.
Memory objects in dream bubbles have so far shown little to no difference with their real counterparts. The memory version of the Pop-a-matic Vrillyhoo Hammer worked exactly the same as the real hammer likely would. A hat created by a roll of that hammer left the bubble temporarily, but may have fallen back into the bubble because it is tied to the bubble. The reason is still uncertain, and probably irrelevant.
A conversation between Vriska and the instance of John slain by Typheus implies that the presence of formely-living entities (in this example John's dad, whose state of life in the doomed timeline was not shown) that are not directly tied with a specific session, even players whose instances in the corresponding universe have been assigned the role of ancestors, are merely illusions - their souls will not be taken into the dream bubbles upon their demise and no other places of afterlife have been seen, although Aradia did summon the souls of Vriska's victims to torment her in retribution for crippling Tavros.
It is hard, as such, to determine how real memories can become, i.e., if Equius' memory of Aurthour is actually Aurthour's soul, or just a very interactable representation (as is only guessable with the two other lusii seen in the bubbles (Crabdad and Sollux/Mituna's Lusus)). Rufioh also expresses uncertainty as to whether the instance of his lusus rescued by Horuss is just a memory-based construct.
Given that the Felt's jujus, a bowl of black licorice scottie dogs, and the bent horse hitcher have appeared in Openbound, Spades Slick may have access to the dream bubbles while he sleeps, though these objects may be memories of Andrew just as well. Jack Noir also affected a dream bubble, but he physically entered it rather than dreaming in it. If carapacians indeed dream in the bubbles, this could explain how Vriska could enter WV's dream by taking a nap.
This could also imply that other constructs of Sburb/Sgrub appear in dream bubbles. This means that Leprachauns may also be able to dream in the bubbles. It may also suggest, if they are further explored, that other Carapacians may appear in the bubbles, such as Snowman and other various dead exiles.
It may also act as a resting place for "dead" sprites - the Lusussprites and the sprites of the original twelve trolls.