I don't like playing board games. I used to like playing them (Chess, Risk, Scrabble, Monopoly, Tactics II) but for some reason I don't understand, I became bored with them in my early 20s. So I'm surprised that I like playing Carcassonne so much. (I should make it clear that I've never actually played the physical board game of Carcassonne; I've only played the iOS version of the game on my iPad and iPhone.
I bought Carcassonne for my iPad ($(removed) and it runs on the iPhone and the iPad) over a month ago, and my wife, our seven-year-old daughter, and I play it together at least three evenings a week, and we haven't grown tired of it yet. The object of the game is to take ownership of villages, roads, and pastures that you piece together with tiles randomly drawn (the are 70 tiles in the game). The rules are simple enough that my seven-year-old caught on immediately (it helps her, and my wife and I, that the iOS version doesn't let you place a tile in a forbidden spot, and that it automatically keeps score). The graphics are beautiful, the gameplay is smooth, and it has a nice guitar soundtrack that has a Pavlovian effect on us when we play it. A typical game lasts about 30 minutes, which is the perfect amount of time for a board game. It also seems to be bug free – not once has it crashed on us, which is pretty remarkable, considering most iPad games I've played crash from time to time.
A couple of friends have told me that boardgames made in the last 20 years or so are much more fun than the ones that I used to play, and that I ought to give them a try. They recommend Ticket to Ride and the Settlers of Catan. I just bought Catan HD for the iPad but haven't played it yet. I'll wait for Ticket to Ride to show up on the iTunes store before I try it.