Update: My current suggestion is Networks: A Very Short Introduction as a better first book on complexity. After that one, this one should make a good deal more sense, though it falls short of ideal. I've listed a number of other suggestions below this review.
I can't really recommend this as a first introduction to complexity science. For that, I would highly recommend Steven Johnson's book Emergence. It is not technical. And it's exciting and a breeze to read, aside from all the fun ideas that'll latch into your brain and keep you thinking for long after. You'll definitely walk away with an intuitive grasp of what complexity is all about, though you will not be a practitioner (yet).
As for this book, which I did actually really enjoy and get a number of insights from, I'll have to expand my review later, but here's a start. John Holland was a genius who was ahead of his time even compared to most of the founders of the field described in Waldrop's Complexity. You've heard about his work if you've heard of genetic algorithms, as he invented/discovered those. He also helped design IBM's CPU instruction set and was one of the earliest researchers to experiment with neural networks and artificial ecosystems. This little volume, effectively 90 pages, is a sort of gamble. It's written at too high a level for a beginner, and its focus on Holland's own genre of complex system modeling, with the terminology he personally liked to use, feels more arbitrary than it actually is. Reading this too fast is an exercise in futility; it'll start to sound repetitive. He tries to avoid jargon and math, but the result is that often what he's saying doesn't feel grounded. 90 pages just isn't a lot of space, especially for a field that, as of the writing of this volume, did not have a unified theory. So it isn't like he could just parade that out.
But this does not mean the book is a bad idea. What he's presenting is what's sometimes called an "idea model"—actually several that are connected. Idea models are not complete pictures or theories. They are toy examples that can illustrate how something tends to work. His "bucket brigade" auction of if-then rules (which can quite literally evolve, at least in theory, any kind of process from scratch on a computer) and his related Markov (statistical) process that diffuses those rules among "gated urns" (which can simulate and help you understand the formation of functional units on multiple levels seen interacting everywhere around you in nature, eg, organelles, organs, organisms, and superorganisms—or similarly atoms, molecules, objects, planets, solar systems, galaxies) seem too specific, and then the attached discussion seems too vague and hand-wavy. Working through it can feel a bit like playing an alternate-reality game—it's almost as if you're falling down a fictional world's "rabbit hole" and deciphering clues. But he leaves a trail for you, and I do not believe he is trying to be abstruse at all. It just isn't a total success. For the effort, though, you'll be taught by someone who revealed and clarified the basic ideas of complexity to many scientists at the first meetings of the Santa Fe Institute. Re-reading bits can pull those words out of the haze and attach sense to them.
It was pretty much perfect for where I am. Later I'll list a few more books that would be good starting points (edit: done!), and more details to suggest who would benefit most from this one (edit: attempted, but still thinking about that).
Note: the VSI series has books on Fractals, Chaos, Networks, The Computer, Information, Probability, Evolution, Systems Biology, and other areas that figure into complexity, which happens to be extremely multidisciplinary. This book does seem to avoid overlapping with those too much. When combined with "90 pages" and "no unified theory yet," "multidisciplinary" and "avoiding overlap" may begin to account for the feeling of many readers that its focus is narrower than expected.
—For those looking to get into complexity—
Here are some more suitable books to consider as starting points, the best ones I know:
Emergence (Johnson) - The first book I'd recommend for most readers. May remind you of Wired magazine, but personally I liked that. Written well for a popular audience.
Growing Artificial Societies (Axtell & Epstein) - Hands-on simulation design. Starting at ground level, it'll patiently walk you through how to consider a social phenomenon of interest and create your own rules to simulate it. Though assigned in the second rather than the first class I took, it's a terrific intro to agent-based modeling, the flagship simulation technique we use.
Complexity (Waldrop) - Awesome journalism about the field's founders and the sleuthy thrill of their discoveries. After reading this, you can search for complexity books online and feel like you personally know many of the authors and what makes them tick.
Micromotives and Macrobehavior (Schelling) - Surprisingly smooth, pleasant read by a Nobel Laureate in economics who won partly on account of this book. Introduced the idea of "tipping points" to readers in 1978. Discusses how personal interactions affect neighborhoods and lead to urban decay, gentrification, and segregation.
Complex Adaptive Systems (Miller & Page) - Highly respected introductory guide, mildly academic tone but nicely written. Packs a punch with central concepts, lots to think about, also practical advice on designing simulations. This was the most useful text in the first course I took... can't go wrong with it unless "academic" scares you.
Complexity: A Guided Tour (Mitchell) - The full panorama! Probably the best and gentlest book for a beginner wanting more depth/breadth/detail than you'll find in Johnson's Emergence. Assumes no prior knowledge. Happens to be a favorite with big names in the field (for example Holland cites it in the book I'm reviewing). It's relatively new and short and also popular in intro courses. Very likely it's the book you want. (But I'm only reading it now FWIW.)
The Sciences of the Artificial (Simon) - Not recommended as a first book for most people, but I'm including it because it's a complete classic, downright poetic, quotable on every page, with some of my own favorite quotations anywhere (not to mention John Holland quotes it in Complexity: A Very Short Introduction and then brings it up several more times). Imagine an especially deep Carl Sagan book full of giant, timeless thoughts, written by a founder of AI and several other fields. He based it on a key lecture series he gave at MIT in 1968 (then updated several times and expanded with new topics in 1996), but you wouldn't know it started as anything but inspired writing. It's another text from my first course, and I loved it then but found it a bit difficult and didn't finish it for a few years. Basically the genesis of the field—so yes, it has been an introduction for many. Certainly it's dated here and there, but the nice thing about original and accurate thinking is that it never goes out of style. More than a great book on the subject, it's a great book full stop.
Dynamics: The Geometry of Behavior (Abraham, Shaw) - This is kind of a 4-part math comic book, where the "characters" if you will are aspects of chaos theory. It's highly visual, advancing mostly via colorful graphs and their captions. Somewhat difficult to get hold of, but I found it in a local university library, and the first volume was recently reprinted, and the ebooks are available from the author's website. Many scientists got their start in chaos (non-linear dynamics) by reading this series from the 80s. It's a real mind-bender and technically requires almost no math knowledge, though if you haven't taken a calculus course, you'd have to really like math to read the whole thing. Anyway, if this sounds like your cup of tea, it's amazing.
Introduction to the Theory of Complex Systems (Thurner, Hanel, & Klimek) - Not for the faint of heart, but if you have a science/math background, I gather it's the most thorough and complete textbook going. A few people will want to jump straight to this one. (It's also one I'm only just reading now.)
Incidentally, there's almost nothing you can do that's more instructive than simply downloading NetLogo and playing with the models that come with it. In the same way that some people pooh-pooh popular science books as if somehow those just shouldn't exist, some experts pooh-pooh NetLogo as if it should do exactly what they want even if that makes it abstruse and/or highly specific to their endeavor. In reality, NetLogo is incredibly useful, both as a learning tool and as a way to prototype most any complex system you care to name. It is also something of a standard. Put aside any pomposity you hear against it and check it out.
Also, there are good free courses online at ComplexityExplorer. (Currently I'm taking the one on fractals and can recommend it.)
Best of luck!