His most well-known work is his famous Incompleteness Theorem, described as "among the handful of landmark theorems in twentieth century mathematics".  It stunned the mathematical world by proving that in any sufficiently complicated formal system (such as mathematics), there are statements in that formal system which cannot be proved to be either true or false.
His other very important work (equally significant, but less well known) was his work in set theory, where he proved that Georg Cantor's puzzling Continuum Hypothesis was consistent with the Axiom of Choice, and that both were consistent with the Zermelo-Fraenkel axioms. This achievement was characterized as "a tour de force and arguably the greatest achievement of his mathematical life .. because .. virtually all of the technical machinery used in the proof had to be invented ab initio." 
He also did considerable and important work in physics (where he had significant findings in the field of relativity) and in philosophy.
- John W. Dawson, Jr., Logical dilemmas: The Life and Work of Kurt Gödel, Wellesley: A. K. Peters (1997)
- Wang, Hao, Reflections on Kurt Gödel, Cambridge: MIT Press (1987)
- Wang, Hao, A Logical Lourney: From Gödel to Philosophy, Cambridge: MIT Press (1996)
- Amir D. Aczel, The Mystery of the Aleph: Mathematics, the Kabbalah, and the Human Mind, New York: Barnes and Noble (2000) - Although mostly about Cantor and his work on infinities, this fine non-specialist work has a lengthy treatment of Gödel's work in the area.
- Yourgrau, Palle, A World Without Time: The Forgotten Legacy of Gödel and Einstein, New York: Basic Books (2005) - Intended for the non-specialist, this look at Goedel's less-known work in physics also contains extensive biographical material on Gödel.
- S. Feferman, S. Kleene, G. Moore, R. Solovay, and J. van Heijenoort (eds.): Gödel, Kurt, Collected Works. I: Publications 1929–1936, Oxford University Press, Oxford 1986.
- Kurt Gödel at the Stanford Encyclopedia of Philosophy, Introduction
- Kurt Gödel at the Stanford Encyclopedia of Philosophy, Section 2.4.1