First: This is not meant to be complete. These are just a few books I found helpful in understanding yoga.
Second: Reading is a wonderful way to deepen your appreciation of the practice. It takes you on the path of jnana. But it does not substitute for the physical practice. Working with other students, and with experienced teachers is basic. But remember, we are not purely physical, or mental, or emotional, or spiritual. And when we seek to open and stretch different aspects of who we are all the other parts grow as well.
I suggest 2 books to start. One is a book about world religions and one about the particular historical context of the physical practice of yoga as it developed in India.
Paths of Faith; John A Hutchison
I originally read this in its second edition. It is now in its fourth. I think it is helpful, even if you are primarily interested in yoga as exercise (if it can ever be just that) to know something about how and why it developed as it did. This book is not specifically about yoga, or about India. Nor is it a book to tell you what to believe. It is a text book about the world's religions.... history, doctrine, practices. If you have a limited understanding of Buddhism and Hinduism this book gives an excellent and non partisan overview.
Yoga Body: The Origins of Modern Posture Practice; Mark Singleton
This is not a breezy quick read. It is well documented history with footnotes. It is also one of the best books I have read in a long time. I like that the author takes the part of an historian, rather than a practitioner with a particular partisan point of view (do you sense a pattern? I do not want to tell you what to believe, but to suggest tools that will help you figure that out for yourself.)
(You could follow this by reading The Science of Yoga; William J Broad. It is a good book, though I find the journalists insistence on the titillating, and preference for making a good story over carefully organizing details less satisfying than Singleton's book. However, it does a great job of explaining how science and belief can intertwine)
The Yoga Sutras; Patanjali
Unless you read Sanskrit you will have to get this in translation. There are a number of translations available, generally with commentary. Since teachers of yoga have hearkened back to Patanjali for a very long time it is a good place to start. Please note I do not recommend much of what is recommended in these aphorisms. But there is probably no book more important in the history of yoga, and it is worth reading to get a sense of what it was, what it is, and how we have changed and stayed the same.
The Bhagavad Gita
Along with the Sutras one of the most important philosophical sources underlying yoga philosophy.
The Hathayoga Pradipika or The Gheranda Samhita
These books are more for the person who really likes to read original texts for themselves, rather than books that explain them. They are not going to be fun reading for the general yoga student.
I suggest these because they give a good idea of how physical practice was understood in the early days. The Gheranda Samhita was written about three hunderd years ago, the Pradipika was earlier. Again I DO NOT RECOMMEND DOING THE PRACTICES FROM THESE BOOKS. There are a lot of things I do not see as useful, and do see as potentially dangerous. However, they are fascinating, and paint a picture of how yoga developed into what we do today.
Light on Yoga; B K S Iyengar
A very thorough guide to asana practice. There are over 500 postures, with intensely detailed thoughts on alignment and sequencing. However, these are black and white photos of Iyengar himself. They are not made to look like yoga mat ads with beautiful bodied slim muscular bodies glistening with sweat. And the shear size of it can be intimidating. If it seems like it is too much, never fear.... there are dozens of really good books with illustrations or photos and instructions for doing postural yoga. This is very serious, but by no means the only one, or the one with which you need to begin. On the other hand, if you enjoy this book, you could then read Light on Pranayama.
The Heart of Yoga: Developing a Personal Practice: T K V Desikachar
A fascinating and wonderful book. Desikachar was the son of Krishnamacharya. The history part of the book is interesting, but the practical insight of how to create a practice of your own is really useful.
While not the most 'important' book on the list, if you were going to read just one, I might suggest this one, if what you are looking for is a way to integrate a practice into your life
Power Yoga; Beryl Bender Birch
An interesting history from one of the 'first generation' of westerners who studied with Pattabhi Jois. I also put it in because I enjoyed reading something from the perspective of a woman. One of the things about yoga is that men and women did not practice yoga together. In fact women came to practice it later, but now it is interesting that women much dominate men as students in typical western classes.
Yoga Journal's Yoga Basics
If you are just beginning and do not want to go too deep into the historical or philosophical or spiritual, but just get a good basic understanding of the discipline this can be read fairly quickly, and includes both background, and rules of practice.