As a mathematician, Kurt Gödel produced a number of important works which are difficult to understand without some kind of editorial guidance or commentary. One excellent book which deals with his incompleteness theorems (as well as a score of other topics) is Douglas R. Hofstadter’s Gödel, Escher, Bach: An Eternal Golden Braid.
Torkel Franzen’s Gödel’s Theorem: An Incomplete Guide to Its Use and Abuse is a good text explaining Gödel to non-mathematicians, and also follow’s Gödel’s influence on modern science and thinking. Gödel’s paper on Einstein’s field equations is the subject of two books by Palle Yourgrau, Gödel Meets Einstein and A World Without Time.
In German:
1931, "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme," Monatshefte für Mathematik und Physik 38: 173-98.
1932, "Zum intuitionistischen Aussagenkalkül", Anzeiger Akademie der Wissenschaften Wien 69: 65-66.
In English:
1940. The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory. Princeton University Press.
1947. "What is Cantor’s continuum problem?" The American Mathematical Monthly 54: 515-25. Revised version in Paul Benacerraf and Hilary Putnam, eds., 1984 (1964). Philosophy of Mathematics: Selected Readings. Cambridge Univ. Press: 470-85.
In English translation:
Kurt Gödel, 1992. On Formally Undecidable Propositions Of Principia Mathematica And Related Systems, tr. B. Meltzer, with a comprehensive introduction by Richard Braithwaite. Dover reprint of the 1962 Basic Books edition.
Kurt Gödel, 2000. http://www.research.ibm.com/people/h/hirzel/papers/canon00-goedel.pdf On Formally Undecidable Propositions Of Principia Mathematica And Related Systems, tr. Martin Hirzel
Jean van Heijenoort, 1967. A Source Book in Mathematical Logic, 1879-1931. Harvard Univ. Press.
1930. "The completeness of the axioms of the functional calculus of logic," 582-91.
1930. "Some metamathematical results on completeness and consistency," 595-96. Abstract to (1931).
1931. "On formally undecidable propositions of Principia Mathematica and related systems," 596-616.
1931a. "On completeness and consistency," 616-17.
Collected Works: Oxford University Press: New York. Editor-in-chief: Solomon Feferman.
Volume I: Publications 1929-1936 ISBN 0195039645
Volume II: Publications 1938-1974 ISBN 0195039726
Volume III: Unpublished Essays and Lectures ISBN 0195072553
Volume IV: Correspondence, A-G ISBN 0198500734
Volume V: Correspondence, H-Z ISBN 0198500750
Writings about Kurt Gödel and his Work
Time and Causation in Gödel’s Universe by John L. Bell
Gödel, Nagel, Minds and Machines by Solomon Feferman
Reflections on Gödel’s Ontological Argument by Christopher G. Small
An Introduction to Gödel’s Theorems by Peter Smith
Gödel’s Theorems And Theories Of Arithmetic by Peter Smith
Gödel’s Incompleteness Theorems by Raymond M. Smullyan
