You don’t need to worry about this if you’re using a tablet, because Amazon lets you read all your Amazon books online.
Before downloading anything from Amazon you need to sign up, which involves you giving them a name and an email address, and them emailing you a confirmation email. To find the free books, search in the Kindle eBook section and sort by ‘Price: Low to High’. Here’s an example search, looking for books about aliens.
To actually get the books, click the title, and on the new page you’re on click on the blue button that says ‘Buy now with one click’. Turn your Kindle on, press the home button, make sure the wireless is on, select ‘archived items’. You’ll see the things you’ve gotten from Amazon, select one and it will appear on your home screen. Now you can read it.
I’m still new to Amazon, and I haven’t actually read any of the books I’ve downloaded from it. If you know of any good books that are permanently free, say something in the comments.