Indie is generally used to reference work that is done independently, or outside of, traditional commercial channels. There is some debate about the title in its application to publishing, but clearly indie publishing is produced outside of the traditional mainstream of big commercial publishers.
For years big publishers controlled what was available to the public to read and enjoy. The process of getting a book published required the work to be accepted by a big publisher. This often involved an agent since unknown authors rarely contacted publishers directly. You could submit your book, but it would be seen with the lowest level of consideration and did not have much chance of making it through the gauntlet of big publishing. Rare books did.
Another avenue was to become famous for something and then big publishers might move your submission up to the top of the pile for consideration. They might even give you an advance and negotiate a larger-than-normal royalty. While this may not seem fair, publishing books was expensive. When a big publisher made a commitment to a book, it entailed staff, production, distribution and promotion expense. Big money. Big publishers committed to books on which their bean counters anticipated a good return on that investment.
At that time, the only way around big publishing was to go to a small press publisher. Small presses faced the same financial expense but were more likely to take on new or unusual authors. They took only a few and, again, getting your work seen and accepted was rare. The only other avenue for an author was to self-publish, also called vanity publishing. The author put up all the money for a run of the book and took on the tasks for which they did not hire such as promotion and marketing in addition to creating the publication. This was very expensive for the author and out of the reach of most.
Enter the digital age. Book publishing was turned upside down. Now anyone could publish and anyone could distribute at no or relatively low cost. Authors might publish to paper or entirely digitally. As you would expect, the big publishers convulsed. The publishing kingdoms they had created and controlled for so many years were now open to the public. This was achieved through the perfect storm of digital media, the internet, print-on-demand and distributors like Amazon.com.
So now back to the question. What is this new thing called? Publishing on your own retains the same nomenclature of the previous eras of publishing and is often referred to as self-publishing or vanity publishing, especially by big publishing who looks down their nose at authors who publish independently. Big publishing is not happy with how publishing has changed and how it has hurt their business. Many authors have adopted the term used by musicians who produced independently of a major label—indie. It is a modern term that, because of its music and film origins, suggests more than just working outside the mainstream. We think it means it is better.
Commercial big publishing products are often passed through so many professional hands that the freshness and originality of the work gets watered down. Each person in the big publishing gauntlet has their impact on the product to ensure that it meets the requirements of the pre-determined market or assigned budgets. Indie publishing avoids those traps.