What are the best books and lecture notes on category theory?
And when you get bored of reading, let the Catsters take over. (78 videos on Category theory!)
Another book that is more elementary, not requiring any algebraic topology for motivation, and formulating the basics through a question and answer approach is:
An added benefit is that it is written by an expert!
I'm also a fan of Tom Leinster's lecture notes, available on his webpage here. In difficulty level, I would say these are harder than Conceptual Mathematics but easier than Categories and Sheaves, and at a similar level as Categories for the Working Mathematician.
Emily Riehl's recently published book Category theory in context is a fantastic introductory text for those interested in seeing lots of examples of where category theory arises in various mathematical disciplines. Understand the examples from other branches of mathematics requires some mathematical maturity (e.g., a bit of exposure to algebra and topology), but these examples aren't strictly necessary to understand the category theory; even the less advanced reader should have no problem understanding the categorical content of the text. It stresses the importance of representability, an understanding of which is crucial if the reader wants to go on to learn about $ 2 $-categories in the future. It's elegantly written, well-motivated, uses very clear notation, and overall is refreshingly clear in its exposition.
The current version of the text is available at http://www.math.jhu.edu/~eriehl/context.pdf and errata in the published version are being updated. The text is new, so it's not as well-known as other texts, but it's so well-written that it seems very likely that it will soon become a mainstay in the world of category theory texts.
9 July 2017 Edit. Updated the link to the text.
Paolo Aluffi, Algebra: Chapter 0 has category theory woven all through it, particularly in Chapter IX of course. I can tell that randomly sampled pieces of the text are well-written, although I have never systematically read longer parts of it.
The nLab is a great resource for category theory.
The first few chapters of Goldblatt's Topoi: the categorial analysis of logic provide another fairly elementary introduction to the basics of category theory.
I've read a fair amount of Sets for Mathematics and found it to be a gentle introduction.
As a young student, I enjoyed Peter Freyd's fun little book on abelian categories (available online as a TAC Reprint). The nice thing about Freyd's book is it isn't boring, and it has little pieces of wisdom (opinion) such as the remark that categories are not really important, you just define them so you can define functors. And in fact you just define functors so you can define natural transformations, the really interesting things.
Of course you may disagree, but blunt debatable assertions (like this one) always make for more interesting reading. Another provocative remark by this author is the observation that he himself seldom learnt math by reading books, but rather by talking to people.
From the nice link above I learned that Goldblatt also quotes a remark (which may have inspired Freyd's) by Eilenberg and Maclane that categories are entirely secondary to functors and natural transformations, on page 194 where he introduces these latter concepts.
Leinster's notes linked by Patrick, look nice - a bit like an introduction to Maclane's Categories for the working mathematician, chatty and full of debatable assertions, (many of which I disagree with, but enjoy thinking about). He does not give much credit, but I believe the adjoint functor theorems he quotes without proof, (GAFT,...) may be due to Freyd. Leinster's notes are easy reading and informative.
Arbib, Arrows, Structures, and Functors: The Categorical Imperative
More elementary than MacLane.
I don't know very much about this, but some stripes of computer scientist have taken an interest in category theory recently, and there are lecture notes floating around with that orientation.
Steve Awodey has some lecture notes available online too. (Awodey's newish book is expensive, but probably rather good)
Patrick Schultz's answer, and BBischoff's comment on an earlier answer also have good links to freely available resources.
Last year the book Basic Category Theory by Tom Leinster was published by Cambridge University Press. I think it can serve very well as an introduction to Category Theory. It covers much less than Mac Lane's Categories for a working mathematician, but motivates concepts better.
And it's also available on Arxiv.
MATH 4135/5135: Introduction to Category Theory by Peter Selinger
(17pp). Concise course outline. Only wish it covered more topics. Available in PS or PDF format.
Handbook of Categorical Algebra (Encyclopedia of Mathematics and its Applications) by Francis Borceux. Rigorous. Comprehensive. This is NOT free, but you can see the contents/index/excerpts at the publisher's web site, listed below. This is a three volume set:
(v. 1) Basic Category Theory, 364pp. (ISBN-13: 9780521441780)
(v. 2) Categories and Structures, 464pp. (ISBN-13: 9780521441797)
(v. 3) Sheaf Theory, 544pp. (ISBN-13: 9780521441803)
Reprints in Theory and Applications of Categories (TAC). This site has 18 books and articles on category theory in PDF, including several by F.W. Lawvere.
Abstract and Concrete Categories-The Joy of Cats by Jirı Adamek, Horst Herrlich, and George E. Strecker (524pp). Free PDF. Published under the GNU Free Documentation License. Mentioned already by Seamus in reference to Wikipedia's external links for Category Theory, but worth repeating, because it's very readable.
A Gentle Introduction to Category Theory (the calculational approach) by Maarten M. Fokkinga (80pp).
- Appendix of Abstract-Algebra by Dummit & Foote http://www.amazon.com/Abstract-Algebra-Edition-David-Dummit/dp/0471433349
- An introduction to Category theory by Harold Simmons http://www.amazon.com/Introduction-Category-Theory-Harold-Simmons/dp/0521283043/
- A course in Homological algebra - Hilton and Stammbach http://www.amazon.com/Course-Homological-Algebra-Graduate-Mathematics/dp/0387948236/
Barr and Wells, in addition to Toposes, Triples and Theories, have written Category Theory for the Computing Sciences, a comprehensive tome which goes through most of the interesting aspects of category theory with a constant explicit drive to relate everything to computer science whenever possible.
Both books are available online as TAC Reprints.
I'm surprised that this hasn't been mentioned already.
"Category Theory: An Introduction" by Herrlich and Strecker. You can find this book in either the Allyn and Bacon Series in Advanced Mathematics or Sigma Series in Pure Mathematics.
Herrlich and Strecker co-authored another book called "Abstract and Concrete Categories: The Joy of Cats" which is not nearly as good as the former book.
Lawvere, Rosebrugh. Sets for Mathematics.
Pierce B. C. Basic category theory for computer scientists.
José L. Fiadeiro. Categories for Software Engineering.
Martini. Elements of Basic Category Theory.
Burstall, Rydeheard. Computational category theory. Requires ML background.
"Basic category theory"is a script by Jaap van Oosten from Utrecht university (u can find more scripts on topos theory and intuitionism there). Advanced is Introduction in Higher order categorical logic by Lambek & Scott. The 3 vols. from Borceux aswell as Johnstone: Sketches of an elephant, 1-2 are very readable reference for looking up proofs and technical details. Toposes and local set theories by Bell is availlable in Dover prints.
There is also this Introduction to Applied Category Theory Course offered by MIT. The lectures conducted by David Spivak and Brendan Fong are recorded and posted here. I personally find this a much better introductory material than the other one I posted, despite being a software engineer.
A relatively new source tha I believe hasn't been mentioned: Notes on Category Theory with examples from basic mathematics by Paolo Perrone.
These notes were originally developed as lecture notes for a category theory course. They should be well-suited to anyone that wants to learn category theory from scratch and has a scientific mind. There is no need to know advanced mathematics, nor any of the disciplines where category theory is traditionally applied, such as algebraic geometry or theoretical computer science. The only knowledge that is assumed from the reader is linear algebra. All concepts are explained by giving concrete examples from different, non-specialized areas of mathematics (such as basic group theory, graph theory, and probability). Not every example is helpful for every reader, but hopefully every reader can find at least one helpful example per concept. The reader is encouraged to read all the examples, this way they may even learn something new about a different field.
Particular emphasis is given to the Yoneda lemma and its significance, with both intuitive explanations, detailed proofs, and specific examples. Another common theme in these notes is the relationship between categories and directed multigraphs, which is treated in detail. From the applied point of view, this shows why categorical thinking can help whenever some process is taking place on a graph. From the pure math point of view, this can be seen as the 1-dimensional first step into the theory of simplicial sets. Finally, monads and comonads are treated on an equal footing, differently to most literature in which comonads are often overlooked as "just the dual to monads". Theorems, interpretations and concrete examples are given for monads as well as for comonads.
In addition to the resources people already added here, I'll mention the resources on this link which are exclusively the more pedagogic resources, and not aimed necessarily at someone with a mathematical background. These include books and lecture notes, but also blog posts.