One of the best-selling board games to come out of Europe in recent years is Ticket to Ride, a highly acclaimed and award-winning railway-themed game designed for two-to-five players.
Play begins on a map of the United States, with various rail lines spanning the country between different cities. The goal of the game is to link cities on the map to each other using trains. Every completed route nets you points. At the start of the game, you draw "tickets"-- cards that give you specific routes to complete. Completing these routes can score you far more points than ordinary routes will, but if you fail to complete them by the end of the game, you will lose that many points instead. There are only limited ways to get from Point A to Point B, so there is an incentive to get your route complete before the other players get to it first.
I prefer games like Ticket to Ride to more well-known games like Monopoly or Sorry, because all players play until the end of the game, when the points are tallied. Which player is winning is not often clear during gameplay, so this encourages everyone to stay engaged in the game until the very end.
It plays as quick as you like-- I've played a rapid-fire two-player game in 10 minutes, and I've also sat around with a bunch of friends, telling stories while playing casually for hours.
Ticket to Ride is available as a physical board game, but is also on iPad, iPhone and Xbox 360. The digital versions do a phenomenal job of capturing the feeling of the game and are a great way to introduce yourself to Ticket to Ride.