What are good books that introduce causal analysis? I'm thinking of an introduction that both explains the principles of causal analysis and shows how different statistical methods could be used to apply these principles.
Pearl recently published a new book, aimed for beginners: Causal Inference in Statistics: A Primer. If you have never seen causality with directed acyclic graphs before, this is where you should start. And you should do all the study questions of the book —— this will help you get acquainted with the new tools and notation.
Pearl is also releasing a book aimed for the general audience, The Book of Why which will be available May 2018.
Also aimed for beginners, Miguel Hernán has just started a new causal inference course on edX Causal Diagrams: Draw Your Assumptions Before Your Conclusions.
As other people have mentioned, Morgan and Winship is a very good textbook --- for social scientists a very friendly yet comprehensive introduction --- and it covers both graphical models and potential outcomes.
There's a recent book by Imbens and Rubin, which covers to a greater extent some parts of randomized experiments, but there's nothing on DAGS --- it will only expose you to the potential outcomes framework, so you need to supplement it with other books, as the one mentioned above.
Among economists, the graduate and undergraduate books by Angrist and Pischke are popular. But it's important to notice they focus on common strategies/tricks --- instrumental variables, differences-in-differences, RDD etc. So you can get a flavor of a more applied perspective, but with only that you won't get the bigger picture about identification problems.
If you are interested in causal discovery and want a more Machine Learning oriented approach, Peters, Janzing and Scholkopf have a new book out Elements of Causal inference, the pdf is free.
It's worth mentioning here the "Causality in Statistics Education" prize. On its webpage you can find slides and other materials for several classes that won the prize for each year since its beginning on 2013. In this vein is also worth noticing VanderWeele's book.
Finally, as obviously already mentioned, there's Pearl's now classic book. The readings of the more preliminary materials cited above will help you reading it.
Morgan and Winship is particularly good on what must be assumed for causal interpretations of regression-type models.
Pearl (2000) is in no sense introductory, although ultimately a very good read. You may find some of his website and specific articles useful, particularly on interpreting structural equation models. They are mostly available as technical reports.
Update: Pearl, Glymour and Jewell's (2017) Causal Inference in Statistics: A Primer, is introductory though. And very good too.
I have very high expectations for Austin Nichols' forthcoming book Causal Inference: Measuring the Effect of x on y. The expected publication date is 2013. In the mean time, his handout and paper provide a nice overview of panel methods, instrumental variables, propensity score matching/reweighting, and regression discontinuity. The comparisons between all these estimators (and RCTs) are especially useful, as well as the Stata mini-tutorials (that can be skipped if you're not a Stata user). Curated references are provided if you want to dig deeper. Unfortunately, there's not very much on structural equations here, though that is also true of the Morgan and Winship book. Their ARS paper is a shorter, though somewhat dated, overview.
I found Pearl to be an interesting, but difficult, introduction to this material. If it was my first exposure to these ideas, I don't know if I would have walked away after reading it knowing how to apply any of the methods very well.
Finally, here are video presentations and slides by economist James Heckman and Pearl from the 2012 Causal Inference Symposium at University of Michigan. Lots of stuff on structural models here.
Cosma Shalizi's textbook Advanced Data Analysis from an Elementary Point of View has an excellent coverage of causation. (The textbook is still in draft form, and is available online as a pdf, so it has the added benefit of being free.)
You should decide, though, whether you are interested in methods for (a) estimating the size of causal effects, or (b) learning the structure of causal networks (i.e. learning which variables influence which others). There are many references for (a), I think Pearl's Causality is the best. There are few introductory references for (b); I think Cosma's textbook is the best, but it is not comprehensive.
CMU hosted some great introductory talks on causal structure learning in 2013. Richard Scheines presented a tutorial on causal inference using Tetrad, a long and gentle introduction to the basic concepts. Frederick Eberhardt presented All of Causal Discovery, a fast-paced overview of the state of the art. One or both of them may be helpful; Frederick's talk should give you plenty of ideas about where to go next.
Gelman is known to be a great author who describes complex concepts thoroughly.
Also consider his web blog: http://andrewgelman.com/ there are lots of materials about causal inference.
You don't get the full picture of all possible methods, but you'd probably get a very elaborate explanation about what is going on.
PS: Gelman's 8 schools treatment effect analysis became a classic example of bayesian statistics of hierarchical modelling.