Are there other schools for witches and wizards in the rest of the world? In the books they only speak of 3 schools, but what about Asian or American people?
And even more, were they affected by the events happening in Europe?
From Goblet of Fire:
"Spect they go to some foreign school," said Ron. "I know there are others, never met anyone who went to one though. Bill had a pen-friend at a school in Brazil ... this was years and years ago ... and he wanted to go on an exchange trip but Mum and Dad couldn’t afford it. His pen-friend got all offended when he said he wasn’t going and sent him a cursed hat."
There aren't any mentions of actual wizarding schools elsewhere in Potterverse.
It's logical to suppose that they exist, just as schools and Universities exist throughout Muggle world everywhere.
As far as being affected, the Voldemort Wars are mostly (though not 100% exclusively) a British affair - aside from Voldemort hunting for Deathstick and his soul fragment existing in Albania (where he got the diadem from earlier). There's definitely no mention of any schools being affected.
For a general "Are there mentions of non-European wizarding world in HP", see: Why isn't there ever any mention of the Americas in Harry Potter?
Yes, there are wizarding schools outside the UK. Obviously, there's Durmstrang and Beauxbatons, which we hear all about in Goblet of Fire, but there is also the Salem Witches' Institute in the United States:
a group of middle-aged American witches sat gossiping happily beneath a spangled banner stretched between their tents that read: THE SALEM WITCHES' INSTITUTE.
Goblet of Fire - page 82 - US Hardcover
As DVK mentions above, there is apparently at least one wizarding school in South America, in Brazil. And Harry realizes there can't be just one wizarding school (Hogwarts) as he surveys the international crowd at the Quidditch World Cup.
"'Spect they go to some foreign school," said Ron. "I know there are others. Never met anyone who went to one, though. Bill had a penfriend at a school in Brazil ... this was years and years ago ... and he wanted to go on an exchange trip but Mum and Dad couldn't afford it. His penfriend got all offended when he said he wasn't going and sent him a cursed hat. It made his ears shrivel up."
Harry laughed but didn't voice the amazement he felt at hearing about other wizarding schools. He supposed, now that he saw representatives of so many nationalities in the campsite, that he had been stupid never to realize that Hogwarts couldn't be the only one. He glanced at Hermione, who looked utterly unsurprised by the information. No doubt she had run across the news about other wizarding schools in some book or other.
Goblet of Fire - page 85 - US Hardcover
The fact that Hermione was unsurprised at Ron's comment about a wizarding school in Brazil would seem to confirm that there are many foreign wizarding schools, but they are just not mentioned by name.
Asia has a professional Quidditch team, the Japanese team the Toyohashi Tengu, so presumably there could be Quidditch at the school-age level at least in Japan, just as there is with the British teams and Viktor Krum, who plays at the professional level for the Bulgarian National team.
Regarding the Second Voldemort War, Madame Maxime left Beauxbatons for at least a period of time to go with Hagrid to parlay with the giants and her absence from Beauxbatons could have affected the students' morale, just as Dumbledore's absence from Hogwarts (Chamber of Secrets and after his death in Half-Blood Prince) left staff and students uneasy. Igor Karkaroff, Dumstrang's headmaster, abandoned Durmstrang and his students at the end of Goblet of Fire, as he had sold out many of Voldemort's followers to the Wizengamot and Ministry of Magic following the First Voldemort World. He was subsequently tracked down by the Death Eaters and murdered. When he deserted Durmstrang, undoubtedly the Durmstrang students were left confused and were facing a lot of uncertainty.
ETA: 3.28.14 I just happened upon another wizarding school when I was reading something at the Lexicon: Mahoutokoro in Japan. The Lexicon sources Pottermore for this information.
Regarding the Salem Witches Institute, who cares if the witches holding the sign were middle-aged? Some middle-aged people actually attend school as non-traditional students. Or they could have been (somewhat overzealous) alumni. By overzealous I'm referring to the sign the witches have hung up between their tents. For God's sake, here in the US it's totally common for people in their 40s, 50s, 60s, etc, to paint their faces up in their alma mater's team colors and wear a matching clown suit in public, in a show of school pride. If you don't believe that these people exist, definitely peruse the pictorial triumph that is the People of WalMart site.
Finally, as Keen encouraged me in his comment to mention, J.K. Rowling basically says American witches and wizards have their own school of magic:
Q: Can American kids go to Hogwarts? (Kelly)
A: No, they have their own school. You'll find out in Book 4. Hogwarts just serves Britain and Ireland.
World Exclusive Interview with J.K. Rowling - South West News Service - 8 July 2000
Yes, I do note that JKR does not give the name of the American school.
Yes, there are eleven major schools worldwide.
The Pottermore entry on wizarding schools opens:
There are eleven long-established and prestigious wizarding schools worldwide, all of which are registered with the International Confederation of Wizards. Smaller and less well-regulated institutions have come and gone, are difficult to keep track of, and are rarely registered with the appropriate Ministry (in which case, I cannot vouch for the standard of education they might offer).
I’ve put together a list of what we know about each of the schools. Unfortunately that particular page doesn’t have a complete list, but does say this:
The precise location of each of the following schools is a closely guarded secret. […] As a general rule, magical schools tend to be situated in landlocked, mountainous areas (although there are notable exceptions, as will be seen), as such regions are difficult for Muggles to access, and easier to defend from Dark wizards.
In the order they were introduced to the canon:
UK (somewhere in Scotland): Hogwarts School of Witchcraft and Wizardry.
For the Scotland part, the HP Wikia article cites an interview with JK Rowling (from a book; I can’t find an online version):
Hogwarts. It is the finest school of Witchcraft and Wizardry in the world and is run by the finest Headmaster Hogwarts has ever had, Albus Dumbledore. Logically it had to be set in a secluded place, and pretty soon I settled on Scotland in my mind.
France: Beauxbatons Academy of magic
First introduced as a Triwizard Competitor in Goblet of Fire. According to the Pottermore article, it covers much of western Europe:
Thought to be situated somewhere in the Pyrenees, visitors speak of the breath-taking beauty of a chateau surrounded by formal gardens and lawns created out of the mountainous landscape by magic. Beauxbatons Academy has a preponderance of French students, though Spanish, Portuguese, Dutch, Luxembourgians and Belgians also attend in large numbers (both Beauxbatons and Durmstrang have a larger studentship than Hogwarts).
Scandinavia: the Durmstrang institute.
The other competitor in the Triwizard Tournament. The exact location is unknown:
Although believed to be situated in the far north of Europe, Durmstrang is one of the most secretive of all schools about its whereabouts, so nobody can be quite certain.
The common guess is Scandinavia, based on the uniforms.
Japan: Mahoutokoro School of Magic.
First introduced as part of a competition in Wonderbook: Book of Potions. Quoting from Pottermore:
This ancient Japanese school has the smallest student body of the eleven great wizarding schools.
Africa: Uagadou School of Magic.
Also sent a wizard to the potions competition. Apparently serves all of Africa:
Although Africa has a number of smaller wizarding schools […], there is only one that has stood the test of time (at least a thousand years) and achieved an enviable international reputation: Uagadou. The largest of all wizarding schools, it welcomes students from all over the enormous continent.
Russia: Koldovstoretz School.
Mentioned in the potions competition, and there’s a Pottermore booklet floating around the web that mentions the name:
Students from the Russian school, Koldovstoretz, play a version of Quidditch where they fly on entire, uprooted trees instead of broomsticks.
Brazil: Castelobruxo.
Mentioned in Goblet of Fire – Bill Weasley has a penpal there. The name was only revealed recently, when we also learnt that it serves quite a large area:
The Brazilian school for magic, which takes students from all over South America, may be found hidden deep within the rainforest.
USA: Ilvermorny.
This has been the subject of much speculation, because it seems likely to feature in the upcoming Fantastic Beasts movie. For a long time, it was believed that the Salem Witches' Institute was this school, but it turns out the name is a joke on the Women's Institute.
As yet, we know very little about the American school. JK Rowling has been dropping hints on Twitter, which I've been compiling in another answer – beyond that we don't know much.
Pottermore has just revealed the name of the school, but very little else.
That leaves three completely unknown schools. The map of wizarding schools from Pottermore deliberately omits the four schools (Koldovstoretz is missing):
Looking at the map gives a few ideas for other locations:
Australia – that’s the biggest gap in the map, quite cut off from any of the other schools. JKR has hinted in a tweet that Australians have their own school, but we don’t know what/where.
Greece – I don’t see anywhere servicing eastern Europe, and Greek mythology means it’s the sort of place where there might have been early magic.
China – probably large enough to support a magical school, and there are a few famous wizards who we know came from China.
Mahoutokoro is a wizarding school in Japan. It is mentioned in the wand wood information in Pottermore. Apparently, witches and wizards from this school prize wands made from cherry wood.
On Pottermore J.K Rowling says there are '11' wizarding schools in the world the only ones i know of are Hogwarts(Scotland), Durmstrang(North Scandinavia), Beauxbatons (Pyrenees), and Mahoutokoro(Japan). I hope J.K Rowling will release more information on this subject. I got this information from Pottermore.
As an addition to @alexwlchan's thorough answer, it has recently been confirmed in Pottermore that
As for the name of the three currently unknown schools, it seems that they are being kept secret1, as indicated in the map available at the Celebration of Harry Potter at the Universal Orlando Resort (published in J.K. Rowling's Facebook page):
1: Also note that Koldovstoretz School is not in the map nor mentioned in the new Pottermore page whatsoever.
General
The above three schools are the largest and most well-known in all of Europe; they are connected via a competition known as the Triwizard Tournament, with the winning school receiving a trophy to take back home.
With Hogwarts, the above three schools are connected via the Wizarding Schools Potions Championship, with the winning school receiving a golden cauldron.
In addition to the above, there are five unnamed wizard schools, the locations of which remain undisclosed, except for one in Brazil.
One of these may be the Salem Witches' Institute in the United States of America, but this remains unconfirmed.
Well, other than Hogwarts, there are two that were mentioned in the series, Durmstrang and Beuxbatons. Those definatly arn't the only schools in the Potterverse, but those are the ones I know.