On the net, Gödel's name is variously written Godel and Goedel where the letter ö is unavailable. One also sees Go:del, and some people even use the TeX code for ö. If ö is unavailable, a reasonable spelling is Goedel, the use of oe for ö in such a case being traditional in German.
Gödel proved a series of fundamental results in logic, starting with the completeness theorem for predicate logic, which he proved in his doctoral dissertation in 1929. The incompleteness theorems (1931) are what he is famous for, but his work in set theory and constructivism has also been of great importance.
From 1938 until his death in 1978, Gödel was at the Institute for Advanced Study in Princeton.
Information about Gödel's life and work can be found in the Collected Works (Oxford University Press, ed. Solomon Feferman et al.) and in the book by Hao Wang, Reflections on Kurt Gödel. There is also a recent biography by John Dawson, Logical Dilemma: The Life and Work of Kurt Gödel.