These books are all good non-technical works on Gödel's theorem (aka Gödel's incompleteness theorem). If you want to know more about it, you could do a lot worse than read any of them. In fact, you could well read them all, and get different viewpoints on the philosophical meanings different logicians attach to the results.
- Gödel's Proof, by Ernest Nagel
- A short, concise work. If you want to know more about the celebrated theorem in a hurry, this should be your choice!
- (of course) Gödel, Escher, Bach: An Eternal Golden Braid, by Douglas Höfstadter
- Probably the best known. The technical details are a bit idiosyncratic, but still good. Very long, and covers a lot more than just the theorem. Far reaching claims about the nature of consciousness, music, AI, and almost everything else are made; I feel that many of these claims are unsupported. However, it's still an excellent book, if you don't treat Höfstadter's opinions as mathematical results. Makes light of the technical difficulties regarding Gödel encoding (to get all of Gödel and Turing's results, you need to show that encoding a proof is primitive recursive, not just recursive). But doesn't actually say anything wrong...
- Forever Undecided: A Puzzle Guide to Gödel, by Raymond Smullyan
- In Smullyan's usual style, this book is merely a sequence of logical puzzles, with some interspersed explanations. An exciting "hands-on" proof of the theorem. Of necessity, some technical details are omitted, but the concepts of what can and cannot be proved are well presented.
- The Emperor's New Mind, by Roger Penrose
- The most controversial of my recommendations. This book uses Gödel's theorem and some quantum mechanics (Penrose is one of the world's greatest physicists!) to "prove" that AI is impossible. While personally I agree with the result, I don't think Penrose does a good job of proving it. However, the technical explanations of the theorem and related results is strong (and correct).