Contrary to what most people think, while the Android Market is approval free, and basically anyone can release an app for it, Amazon has a review process, similar to Apple's, and can be selective of what apps they allow into their marketplace.
As you might expect, since the Kindle Fire is from Amazon, it has a direct link to a Kindle Fire-optimized version of the Amazon App Store. While this makes it nice and easy to discover and download Apps that will work appropriately on the Kindle Fire, there are still many other apps that are not quite optimized or even approved on the App Store.
So, what if you want to download a non-approved app onto your Kindle Fire? Simply set your Kindle Fire to allow the download and installation of unknown sources, and then find another App Store. Here's how you do it:
Step 1: Go to Settings / Device / Allow Installation of Applications From Unknown Sources
Step 2: Find another App Store. There are plenty of App Stores out there, and once you perform Step 1, they're all available for the browsing and downloading. I've detailed a few below:
The only downside to using one of these other App Stores is that you run the risk of downloading malware that's sitting in disguise as a real app. Other than that, you can have your pick of Apps for your new Kindle Fire.