Amazon hates it when you get ebooks from sites other than Amazon.com because it means they aren't getting paid. So, while it is possible to load books onto your Kindle device or read them in the Amazon Kindle for Android App or Amazon Kindle for iOS app, they make sure it is klunky so that you will prefer to buy books through Amazon.com.
The Kindle app does not look in the Downloads directory for ebooks. So if you download an ebook, the Kindle app won't see it! Even if you use the file manager to open the .mobi file. Instead you must move the .mobi file to the Books or kindle directory on Android or the Documents directory on iOS.
There are two ways to get the file into the Books, kindle, or Documents directory. One option is to use a file manager on your Kindle/Android device to move the file:
Another option is to download the file to your laptop or desktop computer and transfer the file over USB. This page has instructions for the Kindle, Android, and iOS:
For more instructions search for 'sideload kindle' on your favorite search engine.