How To Save An App On A Kindle Fire

The Kindle Fire  is basically a customised Android device. You are essentially saving a website bookmark when you "download" an App. 

When you are on the website, you can see a small bookmark icon on your right side of the bottom ribbon (if you aren't seeing the bottom ribbon, try tapping the page). That will take you to a screen where you will see the site with a big plus on it. Tap that, and it will bookmark the site and create a "favorite".

The current website you have open will also appear on your Carousel on the homescreen.  If you long press (hold your finger tip on it for about a second) the site on the Carousel, you'll see a choice to add it to Favorites. That will keep it available to you on the lower bookshelves on that homescreen, and you'll be able to launch it from there. You can long press it on those bottom shelves to remove it from Favorites.