Settlers of Catan is by far my favorite board game. We encountered it a few years back and it continues to remain the favorite. The standard version is for 4 people, with the minimum age around 10. Extensions are available, and you can easily change some rules if there are only 3 of you. A typical game lasts about an hour.
The players compete to acquire resources and construct roads, settlements, cities etc. There is also an amount of chance since you use two dice at each turn to determine who gets what resources. In that respect it is like many other board games. But there are three main factors that make it very interesting and why everyone feels that they can have a chance of winning each game: (1) the board setup is random, so that everytime you play, it IS a new game (there is a Nexus One app available that allows you to get a more balanced board), and (2) you can trade resources, and only the player whose turn it is can trade so that each time you have a possibility of acquiring points, and (3) there are many ways of acquiring points to take you to victory.
Incidently, Amir Khan once said on his blog that he likes to play Settlers of Catan a lot.
Due to the large number of possibilities the first time you play it seems a bit daunting. But it is worth every bit of you time and effort. The game was apparently invented in Germany to teach about capitalism. It has one the 'Spiel des Jahres' (game of the year) award in 1995.