Good "history of mathematical ideas" book? All too often, mathematical history books include far too much material on the private lives of the personalities involved and not enough information on the actual ideas. Mathematics is a living subject in itself, which is not enhanced by knowing about the practitioners themselves (unless it can be shown how their lives link to their ideas, which is far too complex, speculative, and rarely as successful for shedding light on the ideas, as would a direct analysis of how their new idea grew from previous ones). Besides, can we really claim to know the details of a person's life enough to be able to draw inferences on why they did something? This is why I'm looking for a good history of maths book that focuses on how the ideas developed through time, also including how (and ideally why) the notation changed, why the new ideas were introduced, and so on. In fact, this isn't too hard, as Lagrange admirably demonstrates in his "lectures on elementary mathematics" with his short and insightful exposition on the development of logarithms, where he ends it by remarking that:

"Since the calculation of logarithms is now a thing of the past,
  except in isolated instances, it may be thought that the details [i.e.
  the history/development of the theory of logs] into which we have here
  entered are devoid of value. We may, however, justly be curious to
  know the trying and tortuous paths which the great inventors have
  trodden, the different steps which they have taken to attain their
  goal, and the extent to which we are indebted to these veritable
  benefactors of the human race. Such knowledge, moreover, is not matter
  of idle curiosity. It can afford us guidance in similar inquiries and
  sheds an increased light on the subjects with which we are employed."

(Lagrange was known to focus on the history of the ideas involved whenever he wrote a large treatise, such as the excellent history of mechanics that he opens off his Mechanique Analytique with.)
I couldn't sum up the reason for my interest in the history of the development of mathematical ideas any better.
 A: A lot of excellent answers, but I must add Mathematic for the Million by Lancelot Hogben, which considers also societies conception of the world into the different advances of mathematics. Fascinating.
A: The books by William Dunham ("Journey Through Genius", "The Calculus Gallery", "The Mathematical Universe" are the ones I've read personally) show the evolution of some ideas, trying to make the original reasoning understandable with modern(ish) notation.
A: Howard Eves, Great Moments in Mathematics, Vol. 1 & Vol 2. 
I just finished reading this set for the second time. The text originally comes from a course that Eves gave in the math department at the University of Maine (my alma mater). The mathematics is the main thrust, with gritty details and end-of-chapter exercises throughout; the personal-life histories are fascinating and motivating, but not the end of the story. Highly recommended. 
A: Not a generalist book, but I found both
Marcus du Sautoy's The Music of the Primes (tracing the history of the Riemann hypothesis and all the progress in solving it)
and
Simon Singh's The Code Book (going over the history of cryptography from ancient history through the Enigma through RSA through the present)
to be fascinating, well-written, and well-soaked in great stories of mathematics, mathematicians, and understanding of some of the most significant ideas in math.
A: You might be interested in Bourbaki's Elements of the History of Mathematics. I've only leisurely read a fraction of the book, but it does have an interesting perspective.
I think the book is fairly unique in that a lot of the focus is on undergraduate or beyond topics. It feels to me like, when possible, extra attention is paid when the various subjects became or started to become "modern" in some sense. Lots of ideas are traced, with various branchings and convergences highlighted, always with a sample of corresponding mathematicians.
The book is relatively short, less than 300 pages. A lot of material is simply not present (algebraic or differential geometry, for instance), and what is touched upon is generally done so in a terse way (it is Bourbaki, after all). However, as the material is taken from corresponding books in the Elements of Mathematics series and compiled here, it is always given from the perspective of someone with an extremely good grasp on the main ideas and connections, and the "turning points" of a given subject.
Don't expect encyclopedic coverage, but the book is definitely focused on mathematical ideas. 
A: A book I really like is Victor Katz's book - A History of Mathematics 
Another book with well written easily digestible mathematics is Journey Through Genius - William Dunham
A: Excellent suggestions have already been provided. I provide here books specific to  parts of mathematics, and the related history.
The first one is an impressive book on integration theory:


*

*Constitution de la théorie moderne de l’intégration (The building of the modern theory of integration, unfortunately, I do not know a translation in English), Alain Michel, 1992

*Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers, Joseph Mazur, 2014

*The History of Mathematical Proof in Ancient Traditions, K. Chemla (ed.), 2012

*A Short History of Greek Mathematics, James Gow, 1884 (2010)

*A history of chinese mathematics, Jean-Claude Martzloff and S .S. Wilson and J. Gernet and J. Dhombres (ed.), 2006

*$e$: The Story of a Number, Maor, E., 1994

A: Here are three more recommendations the first two being in a similar spirit to that of John Stillwell:


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*Analysis by its history by E. Hairer and G. Wanner. The material is accessible for undergraduates. It contains besides some historical reprints a lot of instructive graphs and figures, which makes the historical development all the more accessible.


*Mathematical Masterpieces - Further Chronicles by the Explorers by A. Knoebel etal. is a continuation of the Chronicles already mentioned.

The next one does not provide a survey of a whole branch of mathematics, but instead tells us about the historical development of the proof of Fermat's last Theorem.


*

*Notes    on Fermat's Last  Theorem written by Alf van der Poorten, a well known expert in number theory. He provides in $17$ lectures the exciting story about Fermat's last Theorem and the historical development of it's proof. It's great too see how seemingly different mathematical disciplines are deeply connected and we meet  famous people like
Euclid of Alexandria, Diophantus of Alexandria, Pierre de Fermat, Leonhard Euler, Joseph Louis Lagrange, Sophie Germain, Carl Friedrich Gauss, Augustin Louis Cauchy, Gabriel Lamé, Peter Gustav Ljeune Dirichlet, Joseph Liouville, Ernst Eduard Kummer, Harry Schultz Vandiver, Gerhard Frey, Kennet A. Ribet and Andrew J. Wiles.

A: Several answers have mentioned some of the more familiar books, but I find most of them unimpressive. I'd guess that one reason why no one has mentioned a definitive study everyone here would recommend is that the subject is simply too vast, and there are simply too many important ideas, as well as too many possible audiences. (I take it that if you're reading Lagrange you aren't going to be satisfied solely by popularizations.) By contrast, the definitive history of quantum mechanics, hands down, is Mehra and Rechenberg's The Historical Development of Quantum Theory.
In many cases, for careful history you're better off looking for detailed studies, whether books or articles, on particular subjects. There are excellent books on the history of measure and integration, the history of geometry, the history of calculus and analysis, the history of number systems and mathematical notation, and many other subjects. (Some of these are written for an audience with a high level of mathematical maturity.) By contrast, there are remarkably few books written in the past two or three decades that deal with the entire range of the history of mathematics. The more an author takes on, the less room he or she has to explain in detail the intellectual contexts in which new ideas arose and their subsequent development.
Here are three volumes that are more comprehensive than topical.


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*Edna Kramer, The Nature and Growth of Modern Mathematics
This is the single best comprehensive volume I've come across, but very few people seem to know of it. It deserves to be much more widely known. Kramer was a mathematician; she did her PhD at Columbia.


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*Jean Dieudonné, Mathematics: The Music of Reason
This is shorter on the history and longer on the ideas, but still there is plenty of history (Dieudonné wrote many histories on more specialized subjects as well, e.g. algebraic geometry and functional analysis), and you will learn a great deal about the internal development of mathematical ideas.


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*Ivor Grattan-Guinness, Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences
An enyclopedia, but useful, and it will put you in touch with the literature.
A: It's a good point that the biography of a mathematician is largely irrelevant for the subject of mathematical ideas, although there may be exceptions.
David Bressoud writes in an accessible style, including rudimentary biographical remarks in his tracing of the development of the Lebesgue integral (A Radical Approach to Lebesgue's Theory of Integration, Cambridge 2008). There is also a book on the development of the Riemann integral.
Julian Havil's Gamma. Exploring Eulers constant (Princeton 2003), is perhaps the kind of history you are looking for. It demonstrates how awkward the first attempts with logarithms were, and ends with the Riemann hypothesis.
I would also recommend Lakoff & Núñez: Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being. Although this is not a history of mathematics as such, it brings forth some really interesting ideas about how mathematics have developed. They cover the development up to Euler or thereabouts.
A: For some easy reading and good illustrations, I suggest Mathematics An Illustrated History of Numbers. http://www.amazon.com/Mathematics-Illustrated-History-Numbers-Ponderables/dp/0985323043
A: Mathematics: the loss of certainty (Oxford University Press, 1982)
by Kline has long been a favorite of mine. It does a good job of describing the philosophies of mathematics, and impetus for their evolution, throughout history. Though it may be more concerned with the foundations of mathematics than you may be interested in.
A: John Stillwell's Mathematics and Its History is a terrific book for precisely this purpose. See my Amazon review.
I have also written some free history of mathematics course materials myself in a similar spirit.
A: Following @user230452's comment, add some specific numbers, like Maor, "e: The Story of a Number" (Princeton University Press, 1994) and Nahin, "An Imaginary Tale: The Story of i" (Princeton University Press, 2010). I find the second one much better.
Not really "history", but a detailed review (and a jewel in itself) is Artin's "The Gamma Function" (Holt, Rinehart, Winston, 1964)
A: A Concise History of Mathematics

This compact, well-written history — first published in 1948, and now in its fourth revised edition — describes the main trends in the development of all fields of mathematics from the first available records to the middle of the 20th century. Students, researchers, historians, specialists — in short, everyone with an interest in mathematics — will find it engrossing and stimulating.
  Beginning with the ancient Near East, the author traces the ideas and techniques developed in Egypt, Babylonia, China, and Arabia, looking into such manuscripts as the Egyptian Papyrus Rhind, the Ten Classics of China, and the Siddhantas of India. He considers Greek and Roman developments from their beginnings in Ionian rationalism to the fall of Constantinople; covers medieval European ideas and Renaissance trends; analyzes 17th- and 18th-century contributions; and offers an illuminating exposition of 19th century concepts. Every important figure in mathematical history is dealt with — Euclid, Archimedes, Diophantus, Omar Khayyam, Boethius, Fermat, Pascal, Newton, Leibniz, Fourier, Gauss, Riemann, Cantor, and many others.
  For this latest edition, Dr. Struik has both revised and updated the existing text, and also added a new chapter on the mathematics of the first half of the 20th century. Concise coverage is given to set theory, the influence of relativity and quantum theory, tensor calculus, the Lebesgue integral, the calculus of variations, and other important ideas and concepts. The book concludes with the beginnings of the computer era and the seminal work of von Neumann, Turing, Wiener, and others.
  "The author's ability as a first-class historian as well as an able mathematician has enabled him to produce a work which is unquestionably one of the best." — Nature Magazine.

A: Try these books  by Morris Kline:


*

*Mathematical Thought from Ancient to Modern Times

*Mathematics: A Cultural Approach
A: I have taught history of mathematics using the text 
Mathematical Expeditions - Chronicles by the Explorers , Reinhard Laubenbacher and David Pengelley, Springer, 1999, ISBN 978-0-387-98433-9 (softcover). 
(http://www.springer.com/us/book/9780387984346).
Each chapter traces an idea through time with well chosen excerpts from the original literature.
You can see the philosophy  (and some sample free reads) here:
https://www.math.nmsu.edu/~history/
