Please do not attempt to download a Kindle file to your phone or tablet as most of these devices do not give you the ability to save files locally. Instead, please download from a desktop or laptop computer and save it in a place that you can get to later. You'll need the file in step 5.
More information can be found here at this Amazon link.
IF…the book doesn’t show up in your library, your email address hasn’t been whitelisted to send to your Kindle. To whitelist your personal email address, do the following
-----Detailed Instructions with Screenshots-----
1. Visit your Manage your Devices page at Amazon
2. Once you log in, click the Settings tab.
3. Scroll down to the Approved Personal Document E-mail List. If you don’t see your everyday email address, you can add it by clicking the Add a new approved e-mail address.
4. Once your email address is added (or confirmed), scroll up to the previous heading to find your Kindle’s email address.
5. Now you can compose an email to your Kindle from an approved email address and attach the book file as an attachment. You don’t even need to include a subject in your email. Provided that you’ve sent it from your approved address, It should show up in your library in 2-10 minutes depending upon how busy the server is. In the future, you only have to email the file. You won’t have to do steps 1-4 any longer (unless your email address changes).
This not only works for .mobi files as you can email PDFs, Word Documents, and others to your Kindle or Kindle app.