My Favorite Game is Settlers of Catan. In Settlers of Catan players try to be the dominant force on the island of Catan by building settlements, cities, and roads. On each turn dice are rolled to determine the current production on the island. Players collect raw materials to build up their civilizations to gain enough victory points to win the game. This game is a German game of the year winner, primarily because of its amazing ability to appeal to non-gamers and gamers alike.
Also There are 2 expansions; Seafarers of Catan and Cites and Knights of Catan. In Seafarers, players can build shipping lanes, which are very similar to roads. Additionally, the game comes with many different water-hex-heavy variant setups. Cites and Knights adds several new aspects to Settlers of Catan but the two major ones are creating knights to protect the land from invading barbarians and building city improvements that confer benefits upon that city's owner. It also adds tactical complexity to the game and game length. Both expansions are played with the ordinal version of Settlers of Catan and you can play all three together for one big game.
You can play find more on this game at http://www.boardgamegeek.com/game/13. Also you can play it on line at http://catan.jsettlers.org/index.html or http://zone.msn.com/en/catanonline/default.htm or http://games.asobrain.com/