First, Asian players most of all are interested in MMOG's. The market for them is growing rapidly, while in US it is stagnating. I provided some links in the end. While the Western RPG market is dominated by World of Warcraft, Asian MMORPGs are still booming. Worldwide, analysts expect the market to reach revenues of $8 billion in 2010, according to a Strategy Analytics report on BusinessWire.
The U.S. market has been slowing, but not stopped, and Asian market growth has led to an overall global revenue increases. The market exceeded $5 billion in 2009, a 17% growth over the prior year. If the market hits the predicted $8 billion this year, companies will take it as a sign to continue pursuing free-to-play models.
Highest interest goes into free games, or so called Freemium model s, where one can play for free, but get additional benefits by donating some money. The reason for such popularity may be the amount of piracy in Asia, where big companies just refuse to enter the market with single-player games as they are quickly pirated. The freemium model or usual subscription-based model is better suited for the market.
Another big one is the apps in social networks, renren,com , 51.com and kaxin001.com Clones of Farmville with paid virtual items are hugely popular among casual gaming crowd. But here arises the problem of entering the market - you have to find out the details of what is popular and what is not, not even talking about language barrier here, as those networks are traditionally native-languages only.
If we talk about Korea as a separate market, Starcraft is the main player there. There is half a dozen TV-channels streaming games 24/7 there, so you might get the perspective. For additional overview you may check article on wikipedia.