There are a lot of play dough recipes out there and I have tried a few. This one is, so far, my favorite:
Homemade Play Dough
Ingredients:
1 cup flour
1/2 cup salt
2 Tbs cream of tartar
1 cup water
1 Tbs vegetable oil
Mix the flour, salt and cream of tartar in a saucepan. Combine water and oil in a small bowl. Stir into flour mixture gradually. Cook over medium heat for approx. 5 minutes or until very thick, stirring constantly. Remove from heat and allow to cool for a few minutes. Knead until smooth. Add food coloring of choice. Store in Container.
If you would like more detailed directions (with pictures!) Keep reading!
First, take out a sauce pan or stove-top pot of choice and add a cup of flour:
Next, add 1/2 cup of salt:
Then, add 2 tablespoons of cream of tartar and mix all the dry ingredients together:
Now, in a separate bowl, combine 1 cup of water and 1 tablespoon of vegetable oil:
Add the water/oil combination to the flour/salt/cream of tartar mix and stir:
NOTE: I mixed in a little tumeric to give it a yellow color.
Cook over medium heat, stirring CONSTANTLY until you see the mixture start to get really thick:
Now, take it off the heat, let it cool for a few minutes and knead it until it is smooth:
Finally, Let the kids try it out!
It is officially Lemon Drop Approved! 
I also saved the old Play doh containers we had to store our new homemade play dough:
