When you purchase a Kindle book from the Amazon Kindle store, or have a sample sent to your Kindle, or purchase a subscription to a blog or other periodical, these things will be downloaded from Amazon to your Kindle.
While they are downloading: You will see the little connection/downloading indicator in the upper left corner of your Kindle Home screen. When a download is in progress you will see a small animated circle going round and round. If you want to make your Kindle check the Kindle store for anything needing downloading (and so you can see the little animation), press the Menu button, then use the down buttons to navigate to the words Sync & Check for Items. When it is selected (underlined), press in on the main navigation button to make it happen. Immediately look for the little rotating circle at the top left of your screen.
While one or more items are downloading, you can monitor the process by pressing Menu and selecting View Downloading Items. This will display a list of any items that are in the process of downloading, and their download progress. If this option is dimmed, it is an indicator downloading is complete and your items are available in Home.
The Home screen pages are where items automatically appear when you purchase them from Amazon. If the download is not successful, or if it is interrupted, your Kindle will try to download them again. You don't need to do a thing.
Once you have something downloaded, you can organize it even more by creating Collections, and putting your books there.