There are many books which mention ECC. Which one should you read?
The answer depends on who you are, and what you want to learn.
I have grouped the books into four piles, depending on the reader.
Introductory cryptography books written for Computer Science/Engineering students with a moderate mathematics background
 Cryptography, An Introduction by Nigel Smart. Free at this website. A great little introduction to all aspects of cryptography. Elliptic curves and ECC are briefly discussed. This is a good first choice for learning about cryptography in general, and ECC in particular.
 Cryptography: Theory and Practice by Doug Stinson. This is a very good introduction to all aspects of cryptography, from a relatively mathematical point of view. Elliptic curves are briefly discussed. The book is longer and has more detail than Smart’s book. It is an excellent place to start for anyone with a serious interest in learning ECC.
 Guide to Elliptic Curve Cryptography by Darrel Hankerson, Alfred Menezes, and Scott Vanstone. This book is written for computer scientists, engineers and security professionals who have some basic knowledge of cryptography. It gives a very thorough and detailed presentation of the implementation aspects of ECC. Many efficient algorithms for point multiplication etc are described. The book does not deeply discuss the mathematics behind ECC. This book is highly recommended for cryptographers who want to implement or use ECC in practice.
 Introduction to Modern Cryptography by Katz and Lindell. This is a fine book about theoretical cryptography. It mentions elliptic curves. The focus is on rigorous security proofs, rather than practical cryptosystems. The book is not suitable for beginners in cryptography, but is an excellent text for PhD students in theoretical computer science.
Introductory cryptography books written for Mathematics students
 A Course in Number Theory and Cryptography by Neal Koblitz. This is the best “easy” reference for ECC. It contains a nice presentation of the basic ideas. Unfortunately, the book is too elementary to cover many topics which are needed to understand the research literature in ECC. This book is highly recommended for beginners (e.g. undergraduate students in maths or computer science).
 An introduction to mathematical cryptography by J. Hoffstein, J. Pipher and J. H. Silverman. This book contains a good introduction to all sorts of public key cryptography, including elliptic curves, at an advanced undergraduate level. It covers most of the main topics in cryptography, and would be a suitable alternative to the books of Smart or Koblitz. This book is recommended for students of mathematics who want an introduction to the subject.
Books on elliptic curves and/or ECC for research students

Elliptic Curves and Cryptography by Ian Blake, Gadiel Seroussi and Nigel Smart. This book is useful resource for those readers who have already understood the basic ideas of elliptic curve cryptography. This book discusses many important implementation details, for instance finite field arithmetic and efficient methods for elliptic curve operations. The book also gives a description of the SchoofAtkinElkies point counting algorithm (mainly in the case of characteristic two) and a description of some of the mathematics behind the attacks on the ECDLP.
This book is recommended for researchers in the field.  Elliptic Curves: Number Theory and Cryptography by Lawrence C. Washington. This is a very nice book about the mathematics of elliptic curves. It contains proofs of many of the main theorems needed to understand elliptic curves, but at a slightly more elementary level than, say, Silverman’s book. This book would be an excellent next step after the book of Koblitz mentioned above. I recommend this book to all my students and I have not had any serious complaints yet. There is quite a lot of information about cryptographic aspects, including a discussion of the Weil and Tate pairings. This book is highly recommended for those who want to get a deeper understanding of the mathematics behind ECC.

The Arithmetic of Elliptic Curves (2nd Edition) by Joe Silverman.
Every serious researcher on elliptic curves has this book on their shelf. It is an excellent advanced textbook on the topic. The new edition has an additional chapter on algorithms for elliptic curves and cryptography. It is not the place to learn about how ECC is used in the real world, but is a great textbook for a rigorous development of the
theory of elliptic curves.
Advanced books on ECC
 Advances in Elliptic Curve Cryptography edited by I. Blake, G. Seroussi and N. Smart. This is definitely not an introductory text. Instead it covers recent developments in ECC. The intended reader is someone with a basic knowledge of ECC who wants to learn about the latest research developments without having to read all the papers in the subject. I think it is an excellent book, but I am a contributor so am completely biased. This book is highly recommended for researchers in the field.
 The Handbook of elliptic and hyperelliptic curve cryptography edited by H. Cohen and G. Frey. This is an excellent reference for researchers in the field. I increasingly use this book as a reference, and I increasingly find it useful. This book is highly recommended for experts in cryptography.
 Algorithmic Cryptanalysis by Antoine Joux. This book is about algorithms in cryptanalysis. The focus is mainly on symmetric cryptography, but it contains many topics of peripheral interest to ECC, as well as some discussion of pairings.
 Mathematics of Public Key Cryptography by Steven Galbraith. This is an advanced textbook, mostly about cryptography based on discrete logarithms. I am obviously totally biased, but I think it is pretty good. The book is not yet published, as it is not quite finished, but over half the chapters of the current draft are on the web.
— Steven Galbraith
There are two more references which provide elementary introductions to elliptic curves which I think should be mentioned:
* An Elementary Introduction to Elliptic Curves, part I and II, by L. S. Charlap, D. P. Robbins and R. Coley (1988 and 1990); and
* Elliptic Curves and Their Applications to Cryptography — An Introduction by A. Enge (1999).
The documents by Charlap et al have the advantage of being freely available online.
Thanks for your comment Felix.
You’re right that these references are useful. I consulted the Charlap et al notes when I was starting my research career, and they were helpful.
But they were intentionally left out of my list. I think both references are superseded by Washington’s book. In my opinion, it gives a better introduction to elliptic curves and cryptography, at a similar mathematical level to these references.
On the other hand, we always like free stuff.
— Steven Galbraith