Fisher Price ones are great - especially Puppy Play and all the Storybook ones. Elmo Calls and Cookie Calls where Elmo and Cookie Monster actually FaceTime your child is AWESOME, and the Disney Junior Appisodes app is great, full episodes of all his favorite shows.
DS loves all of the Fisher Price Storybook apps, especially the Read and Sing option
He also loves Eebee Baby and Goodnight Safari
PBS Kids is also great, but you need a wifi connection for it to work right
Recently he's gotten into Peek a boo pets, Cut the Rope, and Fruit Ninja, but that was closer to 2 years, not 1
One of his favorite things is to look through our pictures and watch movies of himself and us.
ETA: Cut the rope has an apptivity that you can buy that has a little Om Nom monster that you move around in the game to eat all of the candy - that's the Cut the Rope game he likes, not the physics one! There is also a hotwheels apptivity that we bought, and he likes it, but the game is too hard for him. He just likes driving the car around.
Re: iPad apps
Fisher price- where's puppy? and the coloring one. DD can color and it has a blank option like Paint!
Thanks for the tip to use pictures and movies!!