We have an iPad, but I think at least some of these apps are on the Kindle Fire too. They are all free (although you can pay to get them without ads, but I never do):
My 2 year old likes to play angry birds. He doesn't play the game 'correctly' - he often flings the birds backwards by mistake, but he thinks it's hysterical no matter which way they fly.
I also did a search for "free toddler" and got a shape sorting games that my guy likes to play. It's simply called "Shapes" and by the developer ToddlerTeasers.
There are also books that read themselves to the child and have games incorporated - very cute. The ones I have are by the developer TabTale, I have one called "Dog Story" and "Xmas Tale". This developer has a bunch and they are all free (with ads) or for a small price without ads.
I have one that has pictures of animals and when you tap it, the animals make sounds. It's called Farm Animals by PhotoTouch.
My kids also like Race Penguins and Bubble Birds. I also downloaded one called Sketchpad that kids can just draw with their finger.
Again, you might not find all of these because I have an iPad, but I'm betting you can find a lot of them - or something similar - in kindle format as well.