Hive, based on the game by John Yianni, uses hexagonal tiles in a turn-based, insect-themed two-player game that's a cross between chess and dominoes.
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Board games on the iPhone
iGo replicates the ancient strategy board game with variable board sizes, increasing levels of AI, and multiple game saves.
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Board games on the iPhone
Scrabble also exists on the iPhone in a very good version from EA, but Words With Friends can host many simultaneous games online and has a free version.
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Board games on the iPhone
Klaus Teuber's classic game, The Settlers of Catan, finally comes to life on the iPhone, although without online multiplayer and none of the expansion sets.
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Board games on the iPhone
Simple and very addictive, Mancala is ancient game that plays out like checkers or backgammon. Flipside5 has two versions, one of which is free.