I am currently reading Gian-Carlo Rota's Indiscrete Thoughts. What more can I say apart from "the man can write"? (In other words, you should really read it if you are interested in mathematics.)
I love the insight Rota provides into the "culture" of mathematics. I wonder if there is a similar book that deals less with the cultures and personalities, but more with the "what, why and where?" of the major branches of mathematics as they stand today. Perhaps the book has an entry on functional analysis for instance. Then, taking functional analysis as simply an example, it would have the following information:
What? What is functional analysis?
Why? What sequence of events and motivations led to its birth?
Where? Where is functional analysis today?
Perhaps the book I am thinking of is a much shorter (and dare I say, with a heavy amount of disgust) "chattier" version of the Princeton Companion to Mathematics, that Rota would not have disapproved of? What would you suggest?
As a rule of thumb in order to prune the tree of choices, perhaps we can discount books that have more than a few paragraphs on ancient mathematics.