Popular math books with depth The most wonderful book I have ever read in my life was Fearless Symmetry by Avner Ash and Robert Gross, which is a good book that gives an intuition , and reasons behind the  introducing fields, need for Galois theory etc. 
I am interested in those books which possess the following characteristics ( as possessed by Fearless Symmetry :


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*A  good introduction to the concept, giving the reasons behind introducing theory X or some jargon Y in the arbitrary field chosen.

*Requires a little mathematical background behind understanding that book, must be naive-user-friendly. 

*And must be able to convey the things in a perfect manner.


Any mathematical area is fine with me
To frame in another manner, are there any Analogues of Fearless-Symmetry ? ( in other fields like Algebraic Geometry , Topology....etc) 
 A: For me, I gain a lot of intuition from a book with many well-drawn and colorful figures and then trying to draw my own for the situations they do not present. Two particular books that stand out for me in this regard are the following,


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*Visual Complex Analysis, by Tristan Needham [Amazon Link]

*Discrete And Computational Geometry, by Satyan Devadoss and Joseph O'Rourke [Amazon Link]
Both books have a decent amount of motivation, and in the later book the style is very good for drawing in the reader with pictures and motivation as to why definitions are made and why certain questioned are posed and answered.
If you're looking for a book that gives the motivational explanation behind proofs and specific technical techniques then I would suggest the following,


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*Mathematical Proofs: A Transition to Advanced Mathematics, by Zhang, Chartrand, Polimeni [Amazon Link]


Hopefully those are of the style that you are looking for. Note that these are actual mathematics textbooks, not expository writing.
A: Prime Obsession by John Derbyshire was a fantastic book about the Riemann Hypothesis and the attempts that have been made to solve it. It outlines the history behind the Hypothesis, as well as explaining the incisive reasoning behind the following formula using $n \in \Bbb N$ and prime $p$: $$ \sum_n \frac{1}{n^s} = \prod_p \frac{1}{1 - p^{-s}} $$
The book also brings in the physical connections between the distribution of primes hypothesized by Riemann and the particle physics involved in describing atomic nuclei . . . A fascinating book that isn't afraid to get technical. 
A: For algebraic geometry, I enthusiastically recommend Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra by Cox, Little and O'Shea (Springer UTM series, ISBN-13: 978-0387946801).
The style is that of a textbook, with theorems, proofs, corollaries and examples, and not that of a popular math book, but it's very well written (in a just-casual-enough style that feels right) and formal prerequisites are minimal. The focus on computational aspects of algebraic geometry is what made it particularly interesting and lively.
A: It is hard to ignore (the first half of) Hofstadter's "Gödel, Escher, Bach".  Although there are some problems with the mathematics, it presents the basic ideas behind Gödel's Incompleteness Theorem and outlines the proof.  (The second half of the book is quite good as well, but it focuses on Artificial Intelligence, and the book as a whole can be thought of as a refutation of John Lucas' stance that Gödel's Theorem precludes the possibility of artificial (mechanical) intelligence.)
A: Maybe Geometry and the Imagination by David Hilbert and Stefan Cohn-Vossen.
A: I'd add The Nature of Computation by Moore & Mertens.
A: Weeks, The Shape of Space
Penrose, The Road to Reality
Gowers (ed.), The Princeton Companion to Mathematics
Poston and Stewart, Catastrophe Theory and Its Applications
Courant and Robbins, What is Mathematics?
Lawvere and Schanuel, Conceptual Mathematics
Shafarevich, Basic Notions of Algebra
Alexandroff, Elementary Concepts of Topology
Calculus , Calculus made easy by Silvanus P.Thompson 
Another Fantastic Article that gives a good intuitive start for Algebraic-Geometry is :
Algebraic Geometry by Andreas Gathmann
Colin Adams,  The Knot Book
