In my board game collection (the physical one, not the iPhone one), Carcassonne sits right next to Settlers of Catan as one my favorites. The Xbox 360 has a pretty good version of Carcassonne in its Live Arcade, but the nature of the game--lay down tiles to connect cities, roads, and fields, and settle your little wooden men for points--is a natural for a touch interface, more so even than Catan.
The Carcassonne iPhone app is a beautiful port, with crisp graphics that look exactly like the original board game tiles. There's even a charming old-man narrator who guides you through a surprisingly well-produced tutorial. Games can be played against AI opponents, local players, or over the Internet.
Laying down and spinning tiles to build the board is an easy and well-designed process. Pinching to zoom in and out spreads the landscape out so players can survey the whole game (the board can get quite large as the tiles are laid down in all directions). The developers of Carcassonne promise an update will include an iPad-native version, which current purchasers will be grandfathered in on. Board games are a perfect match for the iPad, even more so for a game that features so many tiles.
A new addition, Solitaire mode, is a clever puzzle mode that's played for points with a set deck of tile cards. Weekly challenges are opened up to friends with new tile sets, and it seems like it could get rather addicting.
For $4.99, a fraction of the board game's cost, Carcassonne's a solid bet. And it's a far more polished game than the frankly ugly version of Catan in the App Store already.