Chess originated in India, and evolved on its spread to the Middle East and southern Europe. The 20th century chess scene was totally dominated by the Soviet Union. But these days, thanks to Viswanathan Anand, one might say that India has finally become the best at its own game.
And the version Anand plays is a Spanish variant of Shatranj from the late 15th century.
The game Reversi or Othello was invented in England, but its modern reinvention was in Japan, and as seen in the link, a number of World Champions in the game have been Japanese. This might be attributed to its similarity with the Asian game Go.
Mahjong is originally a Chinese game, but in America has most of its association with (now) old Jews (mostly women).
Scrabble, a game invented by an American architect, is very popular in Senegal.
Similarly, a Scrabble clone for Android and iOS devices called Wordfeud is absurdly popular in Sweden, with its userbase there estimated at 6 percent of the population.
That Chutes & Ladders board game you remember from when you were a kid? It's actually from India, where it is known as "Snakes & Ladders" and has a much more religious/supernatural flavor (it's intended to teach children about the Hindu concept of karma).
It's still called Snakes and Ladders in some places, even if most of the moral symbolism has been quietly dropped.
Settlers of Catan did okay in Germany and is still one of the best known creations of its author (who was a household name among German board game enthusiasts before that, however), but it is perhaps even more well known and has had more lasting cultural impact in the United States.