twoplayerchess.com website
Two Player Chess is a digital chess board app for tablet and mobile that works great for two people sitting opposite each other - like a traditional game of chess. One person can also play vs the in-built AI which has multiple personalities/strengths, and you can get tips from the AI for best moves in review mode after the game.
Conventional digital chess piece designs are not symmetrical, so if two people are sitting opposite each other then the pieces of one player are upside down! On an iPad/tablet, the pieces are up the right way for player one, however all pieces are upside down for player two.