One thing my family traditionally participated in was the game of Mahjongg. It worked out pretty well because we have four people in our family.
It really doesn't take that much concentration to play, so the mahjongg table is traditionally a place to socialize and just chat. Our family usually had the TV on during dinner, so we generally spent more time talking during the game than at the usual dinner.
At first, mahjongg seems like a somewhat unorganized, noisy game - there is a large pile of tiles in the middle, slamming tiles and shuffling is common.
Mahjongg is a four person game, played with tiles, and is pretty much equivalent to the card game Gin. There are two ways to play the game, a Chinese style and a Taiwanese style. The only difference between the two is calculating points. The game involves making sets of three (or four) making straights or three (or four) of a kind of any tile. There are winds involved (N,S,E,W) and other random characters (a white board, a red "center", and "luck"). The game is as much about building your own hand as it is playing defense in the sense that you lose points if you give the winner his last tile.
There is no real natural (in game) advantage any player has over another, all four players are equal and represents a different 'wind'. On the other hand, there was a large skill differential, my parents being the best (my father a little better than my mother) who would then teach my brother and I; so the order of skill pretty much went in order from oldest to youngest.