Here they are:
Keith Devlin, The Joy of Sets
If you're not a set theorist but want to understand set theory, this book is awesome and one of a kind. I have not read it all, but what I have read I can actually understand!
Frank de Meyer and Edward Igraham, Separable Algebras over Commutative Rings
This classic book explains Galois theory but for commutative rings. Even though there are many more technicalities in the general commutative ring case compared to fields, I actually found the approach in this book more natural than the Galois theory for fields that I learned in undergrad algebra. There are some exercises and this book is easy to read.
T.Y. Lam, Lectures on Modules and Rings
Whoa! Anything you ever need to know about modules is in this book. Not only is Lam an excellent writer, but he provides hundreds of crazy examples of modules that satisfy some properties but not others (like finitely generated flat modules that aren't free). This is the number-one book about modules.
In fact, you should just look at all the books Lam wrote, because they are all great.
Richard Crowell and Ralph Fox, Knot Theory
I've tried to read lots of books in algebraic topology, and most of them I didn't do very well with. You know that feeling where you read and you can go through the proofs line by line and still not really get what's going on? Crowell and Fox's book is not one of those. I learnt a lot from this book, and it will get you computing some knot polynomials and get you to understand the basics of knots. It's one of the few topology books that I actually like!
Bernard Gelbaum, Counterexamples in Analysis
I really like this one. I used it when I took analysis as an undergrad. This is one of the few books that you can just browse around in. It gives all sorts of fascinating examples (like a nowhere differentiable continuous function). Very readable and a must for anyone learning analysis.
Jeffrey Strom, Modern Classical Homotopy Theory
I have been trying to understand homotopy theory for a long time, at least so I can understand some of its applications to algebra (algebra is always my final goal). I'm still working on it, but I feel like I've got the farthest with Strom's book. The cool thing about this book is that it is structured in a problem-solving approach, with much of the text containing little problems as part of the main text.
David Marker, Model Theory: An Introduction
Model theory is a branch of logic that studies mathematical structures from a formal logical point of view. In other words: what can you deduce about groups or rings based on how you can formalize these theories in first-order logic?
I have not found too many books about model theory that are sufficiently motivated for the general mathematician. Chang and Keisler's Model Theory is a classic and a good reference but I can't imagine reading it completely.
Marker's book is one you could read cover to cover, and I consult it whenever I need a refresher in model theory.
Irving Kaplansky, Commutative Rings
There are hundreds of books on commutative algebra. Obviously there's Atiyah and MacDonald, and then there's Eisenbud. Those books have a lot going for them, but I think Kaplansky's book is by far the easiest and shortest to actually understand what commutative ring theory is all about. I have read this one cover to cover and I am supremely glad I did.
Kaplansky has two things going for him in this book that separates it from the other texts in commutative ring theory. The first is that Kaplansky was an unusually good writer. The second is that for the most part, this book reads as a cohesive story in a way. Kaplansky had an excellent eye for choosing just the right topics in just the right depth. In contrast, a lot of commutative algebra texts are a collection of some good stuff you need to know for algebraic geometry. Now that makes an excellent reference for learning algebraic geometry, but it doesn't make a great narrative.
Charles Weibel, An Introduction to Homological Algebra
I do think this is possibly the best text ever written. Every page seems to be full of things you can actually use to do algebra. It has enough abstraction to make sense and yet it teaches you how to calculate as well. After reading this book you can literally do hundreds of calculations that apply to group and ring theory.
Another thing I love about this book is that the proofs make good sense. In a subject like homological algebra, it's easy to get lost in diagram chasing or too many "elementary" arguments, if that makes sense. Weibel introduces just enough heavy machinery so that all the proofs are conceptual and easy to understand and even remember.
Want to help me review these books or others in detail? Try writing a guest post!