Question: How can I download the ebook I just purchased to my Kindle?
The first step (as you've done) is to download the MOBI file you purchased onto your computer.
If you have a USB cable that can connect your Kindle to your computer (note, the new Kindle Fire doesn't come w/ a cable unless you specifically order it), use that to connect. Through your computer, navigate to the "documents" folder on your Kindle (for the Kindle Fire, navigate to the "Books" folder). Copy the .MOBI file to the "documents" folder (or "Books" folder for Kindle fire). Safely disconnect your Kindle from your computer. Your Kindle device will refresh, and the ebook will be listed in your library.
If you don't have a USB to connect your Kindle, then send an email to your Kindle email address w/ the MOBI file attached. If you login to your Amazon.com account, you can go to Manage Your Kindle > Your Kindle Account > Personal Document Settings > Approved Personal Document E-mail List , and set-up approved email accounts from which you can email documents to your Kindle device. Should you require further guidance, I suggest referring to Amazon's online support.
Finally, if you don't have a Kindle device, you can download a free Kindle app for your Mac or PC that you can use to view this file on your computer: