What is normally known as "Gödel's Theorem" (or "Gödel's First Incompleteness Theorem") is the centerpiece of the paper "On Undecidable Propositions of Principia Mathematica and Related Systems" published by Kurt Gödel in 1931. What the theorem shows is that there are statements that can be formulated within the standard axiom system for arithmetic but which cannot be proved true or false within that system. Gödel's paper does this first for the statement "this statement is unprovable", and much of the paper is concerned with showing how such a statement can be encoded within arithmetic. Gödel in effect does this by first converting the statement to one about recursive functions and then—by using tricks of number theory such as the beta function of page 1120—to one purely about arithmetic. (Gödel's main achievement is sometimes characterized as the "arithmetization of metamathematics": the discovery that concepts such as provability related to the processes of mathematics can be represented purely as statements in arithmetic.) (See page 784.)
Gödel originally based his theorem on Peano arithmetic (as discussed in the context of Principia Mathematica), but expected that it would in fact apply to any reasonable formal system for mathematics—and in later years considered that this had been established by thinking about Turing machines. He suggested that his results could be avoided if some form of transfinite hierarchy of formalisms could be used, and appears to have thought that at some level humans and mathematics do this (compare page 1167).
Gödel's 1931 paper came as a great surprise, although the issues it addressed were already widely discussed in the field of mathematical logic. And while the paper is at a technical level rather clear, it has never been easy for typical mathematicians to read. Beginning in the late 1950s its results began to be widely known outside of mathematics, and by the late 1970s Gödel's Theorem and various misstatements of it were often assigned an almost mystical significance. Self-reference was commonly seen as its central feature, and connections with universality and computation were usually missed. And with the belief that humans must somehow have intrinsic access to all truths in mathematics, Gödel's Theorem has been used to argue for example that computers can fundamentally never emulate human thinking.
The picture on page 786 can be viewed as a modern proof of Gödel's Theorem based on Diophantine equations.
In addition to what is usually called Gödel's Theorem, Kurt Gödel established a second incompleteness theorem: that the statement that the axioms of arithmetic are consistent cannot be proved by using those axioms (see page 1168). He also established what is often called the Completeness Theorem for predicate logic (see page 1152)—though here "completeness" is used in a different sense.