Piczle Lines is pretty much the perfect iPhone game. It's a quick, pick-up and play puzzle affair, with a Japanese puzzle style that is easy to learn but gets twistier and tougher as it goes along. It's well-polished, with a cute story mode and extra puzzle packs to play and buy as you go along. And oh yeah, it's free to download and start playing.
You should probably be headed over to the App Store by now, but just in case you want to know more, Piczle Lines has you assembling (very low resolution) "photos" by drawing out various colored lines. The start and end of each line are marked by numbers saying how many line segments the lines take up, so your job is to connect the lines in just the right way, recreating whatever pictures you come up with.
Gameplay is simple and easy -- the app tracks your movements very well, and undoing a line is as simple as tracing it back the way it came. The game starts out very easily, with obvious solutions to the puzzles offered, but it gets tough quickly, as vast line segments must be drawn exactly the right way to match up the numbers on either side. Piczle Lines is a lot of fun, very well made, and the fact that it's a free download (with extra puzzle packs to buy if you happen to complete the main set of 20 puzzles) is just icing on the cake.