Bandy, a sport roughly described as field hockey on ice (with a ball) originated in England, but is today nearly exclusive to the Nordic countries and Russia.
The first game of Baseball as we know it was in America, and it is still popular in the original country with the sport being called "America's National Pastime." However, baseball is also immensely popular in East Asia and Latin America. Italy, Spain, and Australia all have professional leagues in their own nations (although most Australians are unaware theirs even exists). Furthermore, the 2013 World Baseball Classic featured such teams as the Czech Republic, France, Israel, South Africa, The Netherlands (who made it all the way to the semi-final round), and Germany, who hosted the first round.
Basketball was indeed invented in America by a Canadian and remains most popular in the US and Canada, but the sport is immensely popular in much of Europe and parts of Latin America, especially in countries neighboring the US, and in many Asian countries. Particularly, it is the "national sport" in Lithuania.
Likewise the Philippines, being a former American Colony. You can't walk further than a block without seeing a basketball hoop somewhere in the vicinity, every young boy in the country knows how to play, and will actually prefer to play barefoot. The country is obsessed. There was evena documentaryabout it.
Bullfighting probably originated in Rome, but has in modern times been endemic to Spain, Portugal, Southern France, and Latin America.
Curling is a Scottish game, yet anyone who has paid any attention at all to it (even via stereotype) knows that Canada has become the true power and home of the the sport, having dominated international competition for decades. There are an estimated 1.5 million curlers (people who play at least semi-regularly) in the world. Approximately 1.3 million of them are in Canada. And not just in playing. Canada's National Research Council has supported projects involving the sport such as studying methods of controlling condensation on the ice. Curling is Serious Business.
Volleyball is made in America and while it is popular there, it's not top-level. In Brazil however, specially after victorious generations in both indoors and beach volleyball, it's popularity is practically only behind soccer.
Cricket originated from England and is popular in the UK and many Commonwealth nations, but soccer overshadows it as the most popular sport; whereas in India and Sri Lanka, Cricket is the most popular sport in those nations and Cricket stars get near-god-like celebrity status, on par with movie stars and actors. Australians are cricket-mad enough to sometimes see their first victory over England in the sport as part of the birth of their national character. Believe it or not, Australia's treasurer actually wrote an article about this.
Motorcycle speedway was invented in Australia, but is today nearly endemic to Europe, especially Poland.
Rugby originated from England; and is very popular in the UK and other Commonwealth nations, but soccer overshadows it as the most popular sport; whereas in Australia and New Zealand, rugby is basically the national pastime (although it shares that distinction in Australia with Aussie Rules Football).
Although The Beautiful Game was generated in England and is immensely popular in Europe, it was in South America (especially in Argentina and Brazil, long time soccer rivals) which helped make its popularity soar. As a Brazilian saying goes: "The English invented football, the Brazilians perfected it." Its importance in South America is so big that the word was changed to fit Spanish phonetics (fútbol) while Mexico still uses the unmodified phonetic (futból). It has also gained immense popularity in Africa and has recently also become so in Asia, in thanks to its simple rules (other than perhaps the offsides) and the only requirement is a ball and two goalposts (or at least anything that resembles a goalpost like a space between two trashcans or something). However, like anything in life, you can't please everyone, and what prevents it from being truly a "World Game" is that the sport has had difficulty in gaining a following in Oceania (most notably Australia and New Zealand) and North America north of Mexico (especially in Canada and the United States), but even in these countries, soccer is still very popular at least as a recreational and youth sport and their national team is still well rooted upon. Furthermore, the Beautiful Game has been picking up in Australia ever since the national team (the Socceroos) reached the Round of 16 in the 2006 FIFA World Cup.
The word "soccer", an abbreviation of Association Football, is used in the United States, Canada, Australia, Ireland, South Africa, Japan, and several other countries to distinguish the different "footballs". However in Britain, the word's country of origin (no, really), as well as other countries that have soccer... er, football as the number one sport, the word seems to make a lot of people cringe. For some reason though, despite its British origins, the Americans are normally blamed and flamed for the creation of the word.
American Football is very popular in Canada, mainly this is because Canadian Football; the second most popular sport in the country behind Hockey, is basically quite similar to American Football, with some notable differences. Toronto for example, has plenty of Buffalo Bills fans, and the team has even had games played there. Some Canadians have even pushed for NFL teams in Canada; but that isn't likely to happen anytime soon, mainly due to the existence of the Canadian Football League.
In the early 1990s, the Canadian Football League started up four teams in the United States. Three of those teams didn't do very well. The fourth, however, moved into Baltimore, Maryland, which had been football-starved ever since the Baltimore Colts snuck out of town in the middle of the night in 1983. Baltimore welcomed their new CFL team with open arms. The team, which changed names from "Colts" to "CFL Colts" to "CFLs" to "Stallions" due to the NFL calling trademark on the Colts name, was turning a significant profit. When the CFL administration decided to end the USA expansion there was talk of leaving the Stallions. However, the NFL was preparing to relocate the Cleveland Browns into Baltimore as the Ravens, and the CFL decided that trying to compete wasn't worth the risk. The Stallions moved to Montreal and became the second Montreal Alouettes (the original team folded in 1987).
While Ice Hockey was created in Canada and is immensely popular there, it is also popular in Russia, Sweden, Finland, the Czech Republic, and the Northern United States. Not so much in the Southern United States though considering that there is little to no snow in those areas and hockey is obviously popular in places where it snows in the winter, although Los Angeles and Dallas are two exceptions—possibly because those areas have large populations of transplanted Northerners. Both cities (along with LA's neighbor Anaheim) have won the Stanley Cup, and titles also helped Raleigh, North Carolina and Tampa, Florida earn fanbases for their respective teams.
Sumo, a form of folk wrestling, was invented in Japan and is immensely popular there (though the professional scene has suffered recently, particularly in the wake of some nasty scandals), but it has also enjoyed popularity in Mongolia, Eastern Europe, and the Pacific.
As for individuals, German-Vietnamese gymnast Marcel Nguyen won a silver medal in the all-around for Germany and became very popular there as a result. However, he's even more popular in Hong Kong, and he has absolutely no idea why.
Another athlete-specific version: Stan "The Man" Musial was one of the greatest baseball players of the 1940s and 1950s and was very beloved in his time, but faded into national obscurity because he was never a Yankee and because he was toonice. Then he took a trip to Tahiti and Australia with his family two decades after he retired and unwittingly found caches of die-hard Musial fans in both countries.
Football/soccer play-by-play man Ian Darke was at best a second-tier broadcaster in the UK, his native country. In the United States however, he's considered THE voice of the game. He became instantly popular with several of his calls during the 2010 World Cup, became the voice for both the US Men's and Women's National Teams, and is widely credited as a major reason why the game is becoming more popular in the US.
Most of the teams competing in VEX Robotics Competition are from the US, VRC's country of origin, but it also has strongholds in China and other Asian countries, as well as New Zealand.