Updated with video. Transcripts added at the end of this post.
Video guide: how to sideload Android apps in Kindle Fire, Kindle Fire HD and Kindle Fire HDX?
Can I run Android Apps on Kindle Fire tablets?
You may already know that Kindle Fire tablet (Kindle Fire, Kindle Fire HD, Kindle Fire HDX) actually is an Android tablet. The Fire OS 3 in the new Kindle Fire HD, Kindle Fire HDX 7 and Kindle Fire HDX 8.9 actually is based on Android Jelly Bean 4.2.
Of course, Amazon hacked Android so heavily that it is hard to find any traces of Jelly Bean in the UI. But this does not mean you cannot install Android apps, for example, from Google Play Store.
Amazon Appstore for Android has a decent collection of apps that are compatible with most Android devices. Similarly, most Android apps can work on Kindle Fire, Kindle Fire HD and Kindle Fire HDX tablets without any modifications.
So, the answer is very clear. Yes, you can run almost all Android apps in Kindle Fire, Kindle Fire HD and Kindle Fire HDX.
Why sideload Android apps in Kindle Fire, Kindle Fire HD and Kindle Fire HDX?
Currently, Amazon Appstore for Android still has far fewer apps than Google Play store. Sometimes, you may need some apps that are available in Google Play, but are not available in Amazon Appstore.
Because the official way to get apps on Kindle Fire, Kindle Fire HD and Kindle Fire HDX is through Amazon Appstore for Android, you have to sideload apps to your Kindle Fire, Kindle Fire HD and Kindle Fire HDX if the apps are from other sources.