Learning the rules for a new boardgame can be daunting. As much as I liked Tigris and Euphrates, for instance, it took us three full sessions just to get the hang of the game's mechanics (and we weren't playing the newer advanced version).
If you need something that you can get into fast, or that's more suited for younger players, you might want to try Blokus:
In Blokus, each player picks a color and tries to lay as many Tetris-like pieces as possible on the board. Each new piece must touch one of your previous pieces, but only on the diagonals. The early going is deceptively peaceful since there's plenty of room for everyone, but eventually you'll be forced to outmaneuver, wall off, and scrabble with your opponents for precious open space.
My friends and I liked Blokus, since it takes the elegant territory acquisition of Go and dumbs it down to the point where almost anyone can learn (I was able to teach the rules to a twelve-year old kid who didn't speak or read English). Admittedly, the game is a bit shallow, without the variation of a deeper strategy title. For quick hits of vicious boardgame conflict between meatier sessions, though, Blokus is a refreshing choice.