Everyone is going to tell you how great Deep Green’s interface is, but the real beauty to me is that I can actually play chess at a reasonable level with it. The chess engine (it is based on “Tiny”) is strong, and it responds quickly. You can take back moves (all the way back to the beginning), get suggestions from the computer, and move pieces either by dragging or tapping without changing settings.

Deep Green is a fast and strong Chess Game for iPhone.
Flaws? It doesn’t bring up a “check” or “checkmate” indicator. It does, however, vibrate, which was more than enough for me. You can’t play another human over a wireless network; you could hand the phone back and forth. That’s it for flaws.
If you’re interested in trivia, Deep Green was originally available for the Newton, which Apple stopped producing shortly thereafter. It takes up less than 2MB on your phone.
Back in the day I paid almost $100 for my Radio Shack board, and upwards of $30 for Chessmaster. Deep Green costs $4.99 during the introductory period ($7.99 thereafter), and it is fantastic. Go buy.