I've had my iPhone for a little while now. Mainly I use it for communication (SMS, e-mail, Twitter) and for reading whilst on the move (Guardian, Google Reader, Instapaper). Occasionally, however, I'll play a game, and Words With Friends has proved to have the best form of gameplay.
Words With Friends is pretty simple. It's basically Scrabble, but on a different board (for copyright reasons). The wonderful thing about it is that you can play against a real opponent (either random or one of your friends), but that you can make your move entirely within your own time - this can be days after the last move. You get notified when it's your 'go' via an SMS-like message (which uses Apple's Push Notification service, meaning that you don't have to have the app open).
This works so well because it means you can fit the gameplay around your life - I usually take my turns whilst on my daily commute, for instance. It also eliminates the time-pressuredness that most games have, and allows the game to be played far more casually.
The problem is that I now expect every iPhone game to have a Words With Friends style of gameplay, but most of them currently don't. The new Blokus game, for instance (based on the great boardgame, which I also own) allows you to play an opponent via Wi-Fi, Bluetooth or via 3G internet, but you have to both be playing simultaneously, with a count-down timer of 60 seconds per go.
Other games that'd benefit from the same style of casual gameplay would be Monopoly, Risk, Trivial Pursuit, and even Connect 4 (which is usually a fast-paced game).
I'm hoping that the Game Center that Apple have announced for the OS4 version of the iPhone will help make it easier for games to incorporate this type of play-by-post gameplay, as well as making it easier to find friends to play with (which is currently a little awkward in Words With Friends).
In the meantime, if anyone would like to challenge me to either a game of Words With Friends (which I'm pretty good at), or a game of Chess With Friends (which I'm pretty bad at), then my username on both is 'frankieroberto'.