Gödel's incompleteness theorem has a reputation of being mysterious and difficult to understand. In
Gödel's Proof Ernest Nagel and James R Newman give a clear explanation of the basics of the proof. They explain the background to the proof and in particular the search for a way of proving consistency of a system of axioms. They then exhibit a system where such a proof is possible - the propostional calculus. The book continues with chapters on the concept of mapping in mathematics, and on Gödel numbering, finishing off with the proof of the incompleteness theorem itself.
I did have some misgivings about the book. I felt that some examples of proofs from axioms would have helped the reader to understand what was going on. Also the book doesn't mention Gödel's β function, which I consider to be the most innovative part of the proof - how you code for 'and so on..' when your system doesn't include such a concept. In summary, the book would be useful for those wishing to get a clear idea of Gödel's theorem without getting too technical, and to go further than what is given in most popular treatments of the subject, but it doesn't go that much further than such treatments.
See also:Gödel's incompleteness theorem project