I am a big fan of The Room, an iOS app which has gained plenty of accolades here and elsewhere already. One of the most charming things about The Room, I think, is how well it replicates real-world objects on a touchscreen platform. The Room expertly turns the touchscreen interface into a series of mysterious puzzle-boxes, all very fascinating and ingenious to play with.
Interlocked is another app that does that, although without a lot of the extra mystery and setting that The Room adds to things. The app is published by Armor Games, the same company that published the great Kingdom