Here's what Amazon has to say about it:
What is Amazon Author Rank?Amazon Author Rank is based on sales of all your books relative to the sales of other authors. Amazon Author Rank shows how an author's books sell relative to other authors. Like the Billboard charts, lower numbers are better. An author with the Amazon Author Rank of #1 in Mystery & Thrillers is the bestselling author in Mystery & Thrillers on Amazon. Amazon author rank is updated hourly.
What's Included in Amazon Author Rank?It's the same approach we use with the book bestseller list we've had for many years - we look at paid sales of all of an author's books on Amazon.com. It includes books in Kindle, physical and audio formats.
Where will Amazon Author Rank be seen on Amazon.com?An Amazon Author Rank will only appear for authors in the top 100 overall or in the top 100 in a browse category. Amazon Author Rank will appear on book detail pages in the More About the Author widget, on an author's Author Page and, on the Amazon Author Rank page.
Reading an Amazon Author Rank in Author CentralIn Author Central, you'll see your Amazon Author Rank over time. The chart will show multiple blue points and an orange point. The blue points show your best author rank of the day, recorded around midnight, Pacific time. They show only that one point in time, and don't represent an average of your Amazon Author Rank throughout the day, but your best rank during the day. The orange point is a snapshot of your Amazon Author Rank right now. It's taken at the beginning of the current hour, and is updated throughout the day.
My current rank in Science Fiction and Fantasy is about #2000-#2500. Alas, not good enough to show up on any page....