once you have purchased the book from amazon the book will automatically appear in the kindle app when you log back into it and it has performed a wireless sync you will need wireless to be able to access your books though because the kindle app will let you read but doesn't save very well.
I can recommend another alternative there are open source books out there for e readers one of the sites i go to is the guetenburg project
On the Mac or on an iPad/iPhone? On the Mac, you set your account in Preferences. I seem to recall it was pretty intuitive. Once the account is linked to your Amazon account, it should sync automatically. In your Amazon account under "Manage my Kindle" you set up what devices to which you want your new purchases to be sent.
EDIT: Can't see the images, you didn't post them, just the file path on your machine.
I see the problem but don't know how to fix it. I purchased the books on my wife's account, Joan's Amazon, and the Kindle app that I am using is Earle's Kindle. How do I change the name of the app to Joan's Kindle?
When you installed the app, did you point it to Joan's account or Earle's? You have to register the app with the Amazon account you want to use for the books. I didn't have two accounts to worry about, so it just worked. Under preferences there is a "deregister" button that may let you change the app over to her account.