This is a rather unique little puzzle game. It is a lot of fun.
The goal is to ensure that each train reaches its destination safely without crashing by building (i.e. drawing) the track with your finger.
There are helpful tutorials, every time a new puzzle element is added (every couple of levels).
Not sure if you are sold? Then try Trainyard Express. Trainyard Express is not a demo but rather the free prequel to Trainyard.
Trainyard Express features 60 puzzles and Trainyard features 150 puzzles. All 210 puzzles are unique, there are no overlaps.
When I wrote this post, I was looking at version 1.6 for iPhone and iPod Touch. Requires iOS 3.0 or later. Available for less then a EURO / Dollar in the AppStore.
Tried with iOS 4.0 and 4.1 on an iPhone 3GS.