Per Pottermore, there are eleven main wizarding schools. If you don't live in a country with its own school (or within a school's primary intake area), you are probably either home-schooled, or taught by correspondence course. One assumes that Muggle-borns tend to be grabbed by whatever the nearest school is, even if it's several countries away. There are also non-approved schools, which Pottermore gives only a passing mention to.
Given that J.K. Rowling is a clear Hogwarts Exceptionalist, the documents written by her have a certain... incompleteness when it comes to the other schools. That, of course, means there's a huge opportunity here for wild speculation! Everything that follows is either direct from Rowling via the Pottermore link, or sheer invention on my part. ^_^
Hogwarts: (Intake: British Isles) Hogwarts was a compromise which pleased none of the Founders. Hufflepuff originally floated the idea of a central magical location which wizards from all the warring factions of Britain could gather in, but the other three were the ones who insisted on making it a school. Ravenclaw wanted a research establishment, Slytherin wanted to teach practical magic - and Gryffindor wanted a combat academy.
Beauxbatons: (Intake: Europe west of Germany & north of Switzerland) The Triwizard Tournament was originally founded to try and stave off war between Hogwarts and Beauxbatons after the English possessions in France were taken by the French. (The situation was made even more complicated by the fact that Scotland was allied with France, but Hogwarts was mainly English.) It was repeatedly cancelled due to wars and outbreaks of nationalism between the two; the 1792 cancellation was intended to be temporary, but after Napoleon's rise to power in France, it remained in place due to tensions between the schools which amounted to outright hatred.
Durmstrang: (Intake: Scandinavia, Eastern Europe) The reason for the tension between Durmstrang and the other European schools is simple: Durmstrang was in the Eastern Bloc, and fell under Soviet influence (partly due to its hatred of Grindelwald the Wizard Fascist). Igor Karkaroff was hated for many reasons, but one was that he was the first non-Communist head of the school in nearly a century.
Koldovstoretz: (Intake: Russia) Koldovstoretz is known from an ancillary text, but isn't shown on the map of schools. The reason for this is simple: nobody knows where it's gone. It went into hiding during the Russian Revolution of 1917, and has never come back out. (This may sound absurd, but their students play Quidditch on entire trees; they're used to moving large objects around.)
Uagadou: [Intake: all Africa, purportedly] Uagadou is not actually a school: it is a city, the capital of the only independent wizarding nation in the world, that just happens to
contain a school. The fact that this is not acknowledged is due to a long-term, deeply racist trend of diminishing the achievements of African wizards, in an effort to brush over the fact that they were the founders of magic; see for example Rowling's characterization of Uagadou students as unruly and mischievous.
There are actually multiple other large, famous, and ancient African schools - but the International Confederation of Wizards refuses to acknowledge them, for much the same reasons. Supreme Mugwump Akingbade has tried to change this, but been stymied by bureaucratic stubbornness.
Ilvermorny: (Intake: North America) While theoretically drawing from the entire continent, Ilvermorny barely receives any students from outside the MACUSA heartlands in the US north-east and west coast. Canadian parents don't see why their children should be bound by MACUSA laws (such as wand registration), wizards from the US south have been in low-key rebellion against the MACUSA government for centuries, and families from Central America would rather send their kids to Castelobruxo.
Castelobruxo: (Intake: Theoretically South America) Castelobruxo actually gets most of its students from Central America, because of its legacy of conquest. The school was actually founded in Yucatan by the Maya, then taken over by Aztec (Mexica, rather) wizards a couple of hundred years later. When the Spanish came to conquer Central America, Castelobruxo moved south - and established itself in the Amazon by force, demanding that native South American wizards submit to their 'far superior' magic setup. Throw in the later effects of Portuguese colonization, and you end up with a school that's in constant tension with its own local population.
Mahoutokoro: (Intake: Japan) Mahoutokoro is actually an exercise in deceit. Every aspect of it - from 'oh, we love this Quidditch you've taught us' to 'our students fly to school on birds' to 'ooh, cherry wood wands are the
most stereotypical BEST!' - has been designed to trick Westerners into dismissing it as a cute little school that's not worth worrying about. What they're actually up to is anyone's guess.
The Missing Three: There are three other schools that have never been mentioned, and much like Koldovstoretz's exclusion from the map, this is absolutely deliberate. Here's the story:
-The Chinese School, despite being almost as large as Uagadou, has been in a state of feud with the international wizarding community for decades. They openly flout the International Statute of Secrecy, and have been directly involved in multiple revolutions in China, including the Communist Revolution.
-The Tibetan School, which served wizards as far away as India, was destroyed by Chinese soldiers during the conquest of Tibet in 1951. Many of its teachers and students escaped, and have established an educational network across southern Asia. The name is maintained on the list of the Eleven, both in defiance against the Muggles who burned it, and in the hope that it will one day be restored.
-The Australian School is known to exist, because wizards from as far away as South-East Asia and Polynesia claim to be graduates from there. But as far as anyone can make out, it exists in another plane of reality entirely; its graduates all come out as devoted mystics, and just insist that they learned their magic in the Dreamtime.
hS