If you visit China and pay attention to what games people play, you will find several games that you are unlikely to have seen at home, at least not in the form played here. It's not uncommon for people to play these games in public: in parks, street corners or wherever there is free space. Without knowing the game, it's hard to get involved, but if you already know how to play, it's much easier to get involved.
Playing games to learn languages
Playing games is a great way of learning a language, not necessarily because the game itself involves many words you need to know outside the game, but because it will be something you have in common with many Chinese people. If you know how to play their games, they will be surprised and often delighted, even if you only know the basic rules and not really how to play well. This can get you to know native speakers and offer you many opportunities to speak about other things as well. Of course, you can also talk about the game itself in Chinese, but this is much harder.
Classic board games (棋类游戏) in China
Learning to play the classic board games in Chinese requires you to learn a few words, but usually not more than a dozen or so. Most games have a core vocabulary that you have to know, without it, you simply won't be able to play.
In 麻将 (má jiàng) "Mahjong", you have to know what the tiles are called, the numbers one to nine and some other words used while playing.
You also need to know the characters for the four cardinal directions (note that 麻将 isn't a single-player puzzle like it's sometimes presented in the West, it's a four player game similar to many Western card games).
When playing 象棋 (xiàngqí) "Chinese chess (lit. elephant chess)" you don't need to say anything, but you do need to be able to recognise the pieces since they are identical save for the Chinese character written on them.There are only a dozen or so different ones, but you need to know them really well. Don't worry, though, a cheat-sheet and a few games will teach you which is which!
Some games, like 围棋 (wéiqí) – "Go" don't require you to speak at all and the game can be played with people from all over the world without exchanging a single word. However, as I said above, simply playing with Chinese people is a doorway to more social interaction and a better understanding of Chinese culture.
Learning the core vocabulary
As a beginner, this might seem daunting, but it's actually not that hard. Once you learn these words, you will use them all the time and they will become second nature in no time. I have even argued that using 麻将 is an excellent way of learning the numbers because you use them so much. The best way of learning a language in general is to use it, and if you enjoy playing games, I can see no better way of using the language than having fun with native speakers at the same time. Repeating the numbers simply to practice is boring, but while playing a game, you use them hundreds of times without even thinking about it!
Playing modern board games
Apart from the classical board games mentioned above, you can of course also play other board games in Chinese. You can also play card games, computer games and any other kind of game you happen to like.
I think this is an excellent way of learning Chinese, but how suitable it is for your particular level depends on what game you're using. Some games are heavily based on languages and won't be suitable for beginners or even intermediate learners, other games involve no spoken language at all and can't really be used for learning Mandarin, but you still get the social benefits of playing games with natives, though, and you can always talk about the game in Mandarin.
Having the rules of a new game explained to you in Chinese can also be an interesting experience, as well as explaining the rules of a game you already know well for your Chinese friends.