Arcanum
An arcanum (pronounced /ɑrˈkeɪnəm/, plural arcana (/ɑrˈkeɪnə/)) is a distinct system of magic, a separate set of rules for how magic works, including how (or if) spells are cast, how enchantments can be placed or removed, and so on. Virtually every particular about magic can vary by arcanum; there are few, if any, constants to all arcana, though there are some rules of thumb that are true in most, but not all, cases.
Most cosmoi where magic exists have one or more particular arcana associated with them. However, there may also be local arcana that function (or at least are known) only in relatively small areas. Nothing prevents a mage from learning to use more than one arcanum; though (in most worlds) mages usually focus on a particular arcanum, mages who are able to use magic from multiple arcana are far from unknown.
Outside of the field of celemology, the word "arcanum" is sometimes used (especially in the plural) to refer to mysteries or secrets of any sort (magical or otherwise). Because of this, the word "celatum" (pronounced /sɛˈleɪtəm/ or /sɛˈlɑtəm/, plural celata (/sɛˈleɪtə/ or /sɛˈlɑtə/)) has been coined as an unambiguous synonym of arcanum in the celemological sense. However, so far on most worlds it remains little used, and while "celatum" has caught on in some circles, and its does seem to be spreading, the Wongery will for the most part comply with the current majority and favor the word "arcanum" instead.
New arcana
Some arcana are inherent in the physical laws of the cosmos where they exist. It's meaningless to discuss when and how these arcana were created, because they weren't; they have presumably existed as long as the cosmos has, or at least as long as the cosmos has had its current physical laws. However, other arcana do seem to be of more recent origin, and less fundamental to the cosmoi where they exist. Some such arcana were even created by humans and other ellogous beings.
The creation of an arcanum, however, is by no means an easy task. Creating a whole system of magic is a complicated endeavor. The creation of an arcanum can be likened somewhat to creating a new programming language on a computer. It can first be done using "source code" in another language, but once some rudiments of the language have been developed, further refinements can be worked using the new language itself. By the same token, new arcana are usually created using an existing arcanum as a base—the new arcanum itself can be thought of as a fixed hapantic. Once the new arcanum has been created, however, it exists more or less independent of the arcanum that was used to create it (just as a programming language has no further necessary connection with the other language that its first compilers were written in), and can be further extended without reference to that original arcanum.
Using another arcanum as a base is not necessarily the only way to create new arcana. In some cases, it may be possible to manipulate the physical laws of the cosmos more directly somehow, or even to utilize technology to cause some change in the cosmos that makes some form of magic possible. The details of such processes, however, depend very much on the physical laws of the cosmos in question, and in any case the creation of a new arcanum by such means is very rare at best.
Not all arcana are the intentional creations of intelligent beings. Sometimes some sort of strong magical enchantment becomes powerful enough to make other effects possible not anticipated by its creators (if it was created at all, and not the result of spontaneous enchantment), thereby giving rise to a new arcanum without any conscious intervention. Just as arcana that are purposely devised, such spontaneous arcana may become independent of the arcanum responsible for the enchantment that engendered them, and become fully separate magic systems of their own.
Subarcana
Although the divisions between arcana are generally fairly clear cut, often within an arcanum there are further natural subdivisions—forms of magic that, while related enough to be considered part of the same arcanum, still bear significant differences, and can perhaps be learned separately. These subsystems may go by a number of names, depending on the arcanum, but the generic name most used by celemologists is "subarcana". It's not uncommon for a mage to specialize in the magic of a particular subarcanum, though learning multiple subarcana is generally much easier than learning multiple arcana.
In some arcana, further subdivisions can be found within the top-order subarcana, perhaps leading to a detailed taxonomy according to which all spells of the arcana can be categorized. The dream magic of Ses, for instance, lends itself to a well-defined carminical taxonomy. Some celemologists extend the taxonomic system upward as well, grouping related arcana together into superarcana. The four types of elemental magic of Dverelei, for instance, are decidedly separate arcana, but do have some similarities and connections that may be grounds for considering them related.
Between worlds
In general, most effects carry over between worlds within a given cosmos. A particular arcanum may only be well known in one particular world, but it will still function in other worlds, so long as the underlying physical principles on which the arcanum relies still hold. There are exceptions; some arcana may depend on special sources of numen only found in particular locations, and may not function too far from those sources unless the numen can somehow be utilized remotely or fabricated. However, aside from cases like these, an arcanum generally works everywhere within a cosmos.
Between cosmoi, however, is another matter. An arcanum depends on the physical laws and conditions of its cosmos, and will not generally function in another cosmos; a mage who finds himself in a different cosmos than he is used to may find his magical skills useless. This is one hazard of intercosmic travel—an explorer who finds a magical means of traveling to another cosmos is likely to find that the spell, talisman, or paracarminical magic he used to get to the other cosmos has no effect there, and that he must find some entirely different way to get back. However, there are some means of circumventing this. It is generally possible, though not necessarily easy, to emulate one arcanum using another, and by this means some spells and other magics have been developed to allow certain arcana to be used in cosmoi where they wouldn't generally apply. It may even be possible to devise a form of intercosmic travel that automatically grants such an emulated version of the arcanum on arrival, but this is very difficult to do, requiring as it does an intimate knowledge not only of the arcanum used to make the trip, but also of the physical conditions and prevailing arcana in the destination cosmos as well.
Curiously, within certain intercosmic aguiae and compits it seems to be possible to emulate arcana from various cosmoi. They may interact in unexpected ways, however, so mages are still well advised to take care with their magics there. Such systatic systems are known in celemological circles as integrative magic.
Mixing
On most worlds, mages who are skilled with more than one arcanum are extremely rare. Those who do exist, however, sometimes find ways to combine the powers of their arcana to produce unique effects. An effect that is easy to achieve with one arcanum but difficult or impossible with the other may be joined with one that cannot be readily achieved in the first arcanum but is difficult in the second, to produce a combined effect that could not have been done easily, or perhaps at all, with either arcanum alone. Furthermore, the metacelemics of one arcanum might allow modifications in the effects of another that wouldn't be achievable with only the second arcanum. The details of combining arcana, however, depend greatly on the arcana to be combined, and those mages who have discovered the methods of doing this tend to guard their secrets jealously.