This seems like a very worthwhile thread to preserve and document, and a good place to compile what is known about arguably the most influential entities in introducing Fallen London from old London’s ashes.
Given the conversion description as well as the central importance of topic material, further warning on material that people MAY NOT WANT TO SEE strikes one as entirely superfluous. Nevertheless: the following likely contains much that you MAY NOT WANT TO SEE.
Regarding the Masters:
Sara Hysaro’s explanation seems quite succinct and sufficient for many purposes. Consider the collective information presented by this admirable scholar as correct in all its material content to the best of my own knowledge. A slightly wider scope, touching on common knowledge for most Fallen Londoners, can hopefully be offered by coupling this with the other data kept here.
The Masters are supremely secretive, which makes discerning much of value about them a great challenge.
They are humanoid or at least capable of emulating a facsimile of humanity, but almost certainly not human in the usual sense of the word, as suggested by their unusual height and atypical manners. A further personal assumption is that they are homogeneous in kind, if not species.
Since they have been around since the beginning of the Bazaar’s residence in the stolen cities of humanity, they possess at least what the author considers "weak immortality" - that is, immunity from death by the normal aging process of most biology. Possible but personally dismissed as vanishingly unlikely is the idea that they were ONCE human and were changed in nature by the greater powers of the universe, most likely the Bazaar itself or the Judgements. They are distinctly limited in knowledge and power, and their plans can be frustrated by the careful plotting of Fallen Londoners with sufficient acumen and resources.
Both the Masters and the Bazaar itself jealously seek expressions of human love (particularly romantic love), whether in written form or in representations of sentimental value.
The Masters are, at least in present Fallen London, expert in the arts of social manipulation and quite knowledgeable concerning the motivations of human beings. They appear to have some code of honour or at least practical decency, but in a supremely diminished occupation - currying gratitude with them is a protracted affair which will demand a hugely disproportionate amount of time and effort. Their personal estates are used to further their individual influences, and (on a… curious number of occasions) thwart the designs of select members of their peers. The Masters’ internal politicking has room for both direct conflict - such as between Mr. Fires and Mr. Stones - as well as alliances. One of the natural results of the scope of their somewhat nebulous ambitions is that their day-to-day motivations prove difficult to untangle at the best of times, and not infrequently contain apparent contradictions of character.
There is a clear correlation between the Bazaar and its attendant Masters, and the symbol system known to many as the Correspondence. While this relationship is still unclear in nature, the Masters are known to be supremely fluent in the use of the Correspondence, and some investigation into the Masters’ machinations reveals that they even use it for the relatively mundane task of relaying messages between themselves.
Through a personal desire to bring down that demi-cryptid known as the Vake, contact with the peculiar martial and ecclesiastical community of the Sisterhood has been established in the past. In discussion with their number, the suspicion that the Vake is in fact one of the Masters’ number was lent considerable credence. The… ahem, "Mother Belligerent" of the order at one point expressed uttermost frustration, condemning the Vake as a "P–" before pure ire caused her to fall silent. Based on this, the assumed homogeneity of the Masters, and an outstanding personal obsession with mirrors and all that lies behind them, the author posits that some might consider the Masters to be Parabolans. This is purest conjecture.
Regarding the Bazaar:
The Bazaar is an entity or aggregate of entities working in concert that warps the world around it in ways thought impossible by traditional understanding of logic, mathematics, physics, etc. This includes alteration to the geography of London on a somewhat regular basis, but may likewise be the cause of numerous other oddities of the Neath such as the extraordinary resistance to true death exhibited by its inhabitants - this in particular is known to cease affecting Neathy persons if they should migrate to the surface, away from the Bazaar’s influence.
One of the most distinctive features of the Bazaar is that its black spires are decorated with a significant amount of the Correspondence’s script.
The Bazaar’s arrival on Earth (and possibly its capacity to "steal" the cities which have fallen through history) are at least in part facilitated by the "Stone Pigs", which are its engines, or steeds, or couriers, or possibly something not quite represented by any of those terms. These creatures are unambiguously physical presences, and have been encountered under London itself by those brave or foolish enough to descend even below a fallen city.
While the Bazaar is known to be a personality distinct from the Masters, it does not interact with those of Fallen London as directly as its Masters do. This introduces ambiguity to the ultimate source of many peculiar edicts of the Masters; some of their actions are clearly taken on their own behalf, such as Mr. Fires’ constant persecution of anything remotely unionized. However, the many odd taxes and prohibitions of the Bazaar’s economy are almost certainly a result of the Bazaar’s own directives, given their near universal enforcement by the Masters.
Rumor has it that the being known to zailors as Stone and to most others as the Mountain of Light is a child of the Bazaar. This poses some very interesting questions, but also suggests that many of the unusual phenomena of the region known as the Presbyterate - often attributed to the influence of the Mountain of Light - arise due to qualitatively similar forces to those exercised by the Bazaar itself.
Should anything of substance be found in addition to or contradiction of the above, those possessing information of interest are requested to provide whatever possible input they feel appropriate. Likewise, the author will provide further data when appropriate and possible.