When I run across a mathematical term or topic that I don't understand, my first stop typically is  Wikipedia, sometimes Wolfram Mathworld.  But the articles there don't always contain the best writing for my interest.  For example,  I may be left wondering what a "sheaf" is really all about.  What was the motivation to create it?  What is the true essence of it?  I can follow the hyperlinks in Wikipedia and learn that the "elephant has a trunk" from one article and that it has a tail from another article and yet another article may talk about the legs.  But I may still not really get it:  What is the "elephant" (or sheaf or Teichmüller space or...) really all about?  Wikipedia can sometimes be too fragmented and, ironically, too technical.  It tries to be accurate and detailed and founded on authoritative references but it isn't always the best source to get the intuition about something, at least for mathematical topics.<br><br>Some might say you should make the investment and get a book on the topic you want to know more deeply.  Yes, I could buy a book about algebraic geometry.  Indeed, I did.  But, typically, a book requires a pretty big investment of time and focus to work through.  And, without the intuition up front, it isn't clear that the payoff at the end will be worth it.<br><br>The Princeton Companion fills the gap.  The focus is on providing a "feel" for topics ranging from elementary to very advanced:  motivations, simple examples, intuitive exposition.  There is not an emphasis on proofs or completeness.  One aspect I really like is that for any particular topic at random, it typically contains the full range of the topic (from simple to advanced), so that no matter what your current understanding is, you can find a place within the narrative to connect with the material and then go from there.  The authors seem to have taken the view of extreme editing, leaving out as much as they can, so that what they leave in is a kind of distillation of the essence of each topic.  What is left is fairly well integrated.  As a result, reading it is like being on a grand tour of mathematics:  You get to sample the best restaurants, see the most beautiful art, wander the nicest shops, without having to commit to living there full-time.