Magical Society

An order, club, alliance, union, or other organisation comprised of people who can do magic. Names vary: it may be called the Mages' Guild, the Circle of Mages, the Conclave of Sorcerors, the Arcane Brotherhood, or whatever. A given setting may have one magical society, several, or none. Historical esoteric societies of any variety are often portrayed as these in Historical Fantasy or Urban Fantasy fiction.

Sometimes, a magical society exists for the benefit of its members, while other times, it is a structure created to control them (whether for benign purposes or otherwise). It could be both at once - by creating rules for themselves, mages may decrease the extent to which people consider them dangerous. It might be a loose support network which only comes together for specific issues, or it might be a rigid hierarchy that demands unity and obedience. A magical society may be responsible for keeping track of magic users, which might involve genealogy or even breeding programs if it's dealing with a Witch Species. It may also act as the Magic Police. Joining a magical society may be an obligation, a rare honour, or anything in between.

Some magical societies are politically powerful (perhaps even running the country), while others are persecuted. In either case, they might keep their existence a secret. There can be conflict between a magical society and other powers (The Government, The Church...), between the society and people who are trying to practice magic outside it, and between members of the society itself (the issue of whether to allow Black Magic is a popular subject).

A magical society may be the ones behind a Wizarding School, especially if the society only exists because untrained magic-users are dangerous. Even if it doesn't have a school, it could be involved in setting up master-apprentice deals. In some settings, it's impossible to learn magic anywhere but one of these.

If a magical society has a headquarters, there's a fair chance it'll be a tower.

Possibly a kind of Weird Trade Union.

Literature

 * In Discworld, one of the unspoken functions of Unseen University is to keep wizards tangled up in bureaucracy and academic politics, and to ensure that they can live comfortably without having to actually do very much. That way, they're not turning their magic on everyone else, and over the years since the university was founded, wizardry in general has become more sedate and inward-looking. Which makes the world a considerably safer place: it's noted that "there were still quite deep scars in old buildings that showed what happened when you had the other kind of wizard", and that in the old days, "the plural of 'wizard' was 'war'".
 * Witches, by contrast, don't have the same level of organisation. They occasionally have big meetings, and some of them will clump together in groups of three, but there isn't any formal leadership. (Informal leadership exists to an extent - it's noted that Granny Weatherwax is "the most highly-regarded of the leaders they didn't have".) One short story deals with someone trying to set up a committee to make their big meeting more organised - it fails.
 * That said, witches always keep an eye on one another to ensure they don't start cackling.
 * In The Black Magician Trilogy, there's a Magicians' Guild that all magic-users are required to join, since untrained magic-users are a danger to themselves and those around them. The plot of the first book has the protagonist (who doesn't really understand the danger, and just wants to be left alone) being hunted by the Guild for this reason.
 * The Wheel of Time has the Aes Sedai. In theory, long-term membership is not compulsory, but they do insist that anyone with power undertake training, and it's implied that the Aes Sedai are very reluctant to let go of anyone with serious potential. They also crack down on anyone being too obvious about using the One Power outside their ranks, as many people consider "can use the One Power" and "is Aes Sedai" to be the same thing. Over the course of the books, it emerges that they're not actually doing as well as they think - there are plenty of people who would make strong Aes Sedai but who, one way or another, are not signed up.
 * Later, one of the protagonists sets up another group called the Asha'man, which consists of men (while all modern Aes Sedai are women). It works a bit differently.
 * In The Dresden Files, the White Council exists as a sort of union of the world's more powerful human wizards. Besides holding various non-human factions (vampires, etc) at bay, it enforces various laws of magic over both its own members and those of lower levels.
 * Harry Potter has the Wizarding World as a whole. They are organized in a bureaucratic government known as the Ministry of Magic, and the have an extensive schooling system, of which Hogwarts is a part.
 * The Order of the Phoenix is something of a Secret Society within the larger magic community
 * The Witcher has the Brotherhood of Sorcerers, which fell apart, and the Lodge of Sorceresses, a secret organisation created afterwards by a group of women who had belonged to it. Their membership is not particularly large, but they can have a considerable impact on politics, and have reputations as untrustworthy schemers.
 * In The Riftwar Cycle, the Tsurani have the Assembly, which is officially outside the law and can do whatever it wants "for the good of the empire". One of the trilogies makes the struggle between magical and mundane authority a central issue.
 * Robert Heinlein's "Magic Inc." involves an attempt to create a non-profit association that would test and license magicians. It turns out to be a diabolical (literally) plot to take over control of all magic use in the U.S.
 * The Enchanter's Guild in Pratt and DeCamp's "The Mathematics of Magic."
 * The Palace of the Prophets in The Sword of Truth series. It's also suggested that something of this nature existed within the Wizard's Keep at Aydindril, but is no more because there are so few wizards left.
 * Jonathan Strange and Mr. Norrell introduces us to the Learned Society of York Magicians, a group of polite English gentlemen who sit around discussing magical writings and history, and who would never do anything so uncivilized as actually attempting to cast a spell or two. Many such societies exist around England, functioning as nice social clubs... Unfortunately for them, when a "practical" magician finally shows up, he turns out to be of a somewhat anti-social type.
 * The Twenty Palaces Society from Harry Connolly's book series.
 * Labyrinths of Echo had orders formed around every nascent magical tradition. There was Age of the Orders when these were main power players, allying jostling and fighting with others and leaving little real power to the monarchy until Order of the Seven-Leaf allied with the King won the civil war, outlawing all other magical orders and use of magic above the weakest grades, since overdraining of magic in the World's Heart was about to cause The End of the World as We Know It.
 * The Castle in Septimus Heap has a fully-developed Magical Society that exists almost side-by-side from the ordinary government.

Tabletop Games

 * Shadowrun products have had a variety of small magical societies, usually informal groupings of like minded magic users. Some were more formal, such as groups set up and run by megacorps.
 * Mage: The Ascension features the Council of Nine, the ruling body of the nine major magical Traditions, which themselves range from mentalist martial artists to Hermetics to enlightened scientists to pagans to reality hackers to shamans. And then there are the many other Crafts that exist beyond them...
 * It also has the Technocratic Union and its five Conventions - Iteration X (engineers and inventors), the Progenitors (biologists and geneticists), the Syndicate (financiers and economists), Void Engineers (explorers), and the New World Order (masters of information and its control). However, many members of the Union see what they do as Enlightened Science, not magic.
 * Ascension's successor game Mage: The Awakening has the Pentacle Orders on one side, and the Ministries of the Seers of the Throne on the other.
 * Likewise, Wraith: The Oblivion has several Guilds, each of which shows a proclivity towards one of the Arcanoi. Some are ill-favored but looked upon as a necessity (the Pardoners, whose knowledge of Castigate helps keep Shadows in check), some are looked upon with scorn (the Haunters, who use the art of Pandemonium to fuck with reality), and some are outlawed entirely (the Mnemoi, whose knowledge of Mnemonysis allowed them to control memory and resulted in the entire Guild being smashed for hideous crimes).
 * Dungeons and Dragons
 * The Orders of High Sorcery in Dragonlance. Created directly by the three gods of magic, they have almost total authority in all magical matters, and being a practicing wizard of substantial power without joining (and thereby being subject to the Orders' regulations) is a crime punishable by death. There are three Orders, White, Red, and Black, each with their own distinct philosophy in magic; representatives of all three sit on the Conclave, which governs all magical matters on Krynn.
 * Later in the timeline, other magical organizations (such as the Grey Knights of the Thorn, a subgroup of the Dark Knights of Takhisis/Neraka) start springing up which are powerful enough to exist independantly of the Orders, as do other forms of magic-users such as mystics and primal sorcerers whose magic doesn't come from the moon gods and therefore lies outside the Orders' jurisdiction.
 * Veiled Alliance of Dark Sun is mainly the underground opposition to Dragon Kings' rule, but it also serves as pretty much the only source of training in Preserver magic.
 * Forgotten Realms, aside of assorted magocracies and arcane corps, has several magical organizations working as behind-the-throne magocracies, such as Arcane Brotherhood or Witches of Rashemen. There are also frequent attempts to found local wizard guilds and stable schools, such as The Covenant—some ending in all-out magic batles and exploding towers, some running for many generations.

Video Games

 * The Elder Scrolls has the Mages' Guild, which seems to be pretty much what the name says - just a guild for mages. Players can join it if they want.
 * The background features the Psijic Order, a more reclusive and restrictive society that the Guild of Mages and the Necromancers (the loose organisation, not the class of mages) sprung from. They are still around, though less important than in their glory days (way back in the 1st and early 2nd Eras), but it took until Skyrim for them to make an appearance in one of the games.
 * In Dragon Age, the Circle of Magi basically exists because the religious authorities don't trust mages and want them under control. Mages have varying opinions about this - some of them agree wholeheartedly, some of them think it's better than the alternative, and some of them want an end to it. The religious authorities have special warriors floating around the Circle's tower to put down anyone who is too proactive in their membership of the latter category. There is a semi-secret alternative organisation, the Mages' Collective, which attempts to defy the regulation imposed on the Circle while still (sometimes) maintaining its own ethical codes.
 * Warcraft has the Kirin Tor. Not the only magical group as there used to be a rival group in Stormwind and the trolls have their own arcane traditions but definitely the most prominent. When they became an independent faction in the second expansion of World of Warcraft mage player characters (regardless of race) start out with a slightly higher reputation than non-mage characters.
 * There were also the Blood elves, but that was pretty much their Magical Society in of itself. The Night Elves also had a Magical Society, the Highbourne. They returned as an explanation for the Night Elf Mage class in the third expansion pack.
 * The Heroes of Might and Magic games have several magic based factions with each game: I had the Sorceress and Warlock, II added the Wizard and Necromancer, III updated the system by having a magic type hero with each faction, with some factions dedicated to magic more than others (the most prominent being the Tower with the Wizards and Alchemists), IV had each faction with their own specialty magic (save for Might), V went back to how I and II did the heroes, with the Academy, Dungeon, and Necropolis for the Wizards, Warlocks, and Necromancers, respectively.
 * The Nasuverse's Mage Association is both this and a Wizarding School. It's presently at an uncomfortable truce with The Church, at war with vampires, is riddled with inter-branch rivalry - most notably between the 'Three Great Branches' Atlas, Sea of Estray, and Clock Tower - and the headquarters itself, Clock Tower, is peppered with rivalry between factions of noble magi. And that's not counting the actions it takes against individual outsiders and other magical organizations outside of Europe.
 * From Kingdom of Loathing come the League of Chef-Magi. This is the guild that organizes the Pastamancers and Saucerors. It's also where they learn new spells.
 * The Clans of Book of Mages the Dark Times. They're training organizations that define the different types of magic, differing in style of dress, Theme Naming, the appearance of their magic bolts, and in the special spells taught to advanced mages. Also, mages as a whole form a very loose society, bound together by the Great Mage, who rules over all mages, and by the institution of the Book of Mages, a Who's Who that ranks the 100 most powerful mages in the world.
 * Dark Souls has the Dragon College of Vinheim. You only hear about it from lore and meet a couple of members, but it's effectively a city state of magic run by the higher ups at the college. Then there is Seath the Dragon and his Channelers, who you fight in game.