Settlers of Catan is a board game that was once memorably described as "perfect". It's based on a tropical island that you and up to three other players have landed on and are exploring. The objective is to accrue 10 victory points by capturing settlements and building roads.
The game is now available on the iPhone. It lets players compete against other humans or an AI, though there's no wireless multiplayer, so you'll have to pass the phone around to allow multiple players to take part. It lets you follow the exact rule set of the original game.
You can adjust the amount of victory points required to win, as well as launch a "friendly" robber mode. It also includes a bunch of beginners' tutorials for those who aren't terribly familiar with the game.
It costs £3, or $5 in the States, and it's available now in the App Store.