Kurt Gödel is of course best known as a mathematician, and some of the mechanics involved with the proof of his first incompleteness theorem had a direct influence on Alan Turing's development of modern computing. But what does this have to do with philosophy?
Well, most directly, Gödel's work contributed to an ongoing philosophical dialogue about the foundations of mathematics, which you can learn about from our episodes on Bertrand Russell at (though in this latter case math was less of our focus than logic and language) Gottlob Frege. These folks wanted to ground mathematics as certain knowledge by creating formal systems that could use a few simple, intuitively obvious axioms to ground all mathematical truths. Gödel showed that this wasn't really possible: that you could certainly construct such systems and try to find proofs for any given mathematically interesting statement in the language (e.g. one tricky one is Goldbach's conjecture, i.e. the claim that every even integer greater than 2 is the sum of two primes), you can't actually come up with an overall proof for a given formal system that every true statement in the system will be provable.
To get a little more formal here, here are Gödel's two incompleteness theorems (as presented by the Stanford Encyclopedia of Philosophy):
1. Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F.
2. For any consistent system F within which a certain amount of elementary arithmetic can be carried out, the consistency of F cannot be proved in F itself.
Now, these theorems do not apply to all formal systems: the system of first-order logic is provably complete, i.e. you can construct a proof in first-order logic for the completeness of the system, i.e. you can show that every true statement in first-order logic can be generated from application of some sequence of transformation rules from the basic axioms of the system.
More importantly for philosophers trying to derive something more generally applicable from the theorems, they only apply to formal, axiomatic systems. Not to "science" as a whole, not to "religion," not to any list of psychological or social principles, not to the precepts of your favorite philosopher. None of those are axiomatic systems, so you can't use Gödel to ground a general philosophical claim that "there are some things that Man can never prove" or "there are some things that we just know to be true that we can never prove." In fact, even in math, if you find a claim in an axiomatic system that you know to be true but which you don't have the resources in that system to prove, then you can always change the list of axioms: hell, just add that very claim as a new axiom (so it follows trivially in a one-step proof from the axioms of the new system), or in practice, this involves moving to a higher level of mathematics, e.g. by introducing things like set theory or Cantor's infinities.
So what are the philosophical implications of Gödel's work, according to Gödel, who was very familiar not only with his immediate predecessors in the philosophy of mathematics, but with folks like Kant and Husserl?
In the 1951 "Gibbs Lecture," developed as an essay (unpublished) called "Some Basic Theorems on the Foundations of Mathematics and their Implications" (which you can read here along with a lengthy, helpful introduction by George Boolos), Gödel first describes what the implications of his findings are for the axiomatization project:
It makes it impossible that someone should set up a certain well-defined system of axioms and rules and consistently make the following assertion about it: "All of these axiomas and rules I perceive (with mathematical certitude) to be correct, and moreover I believe that they contain all of mathematics." If someone makes such a statement he contradicts himself. For if he perceives the axioms under consisideration to be correct, eh also perceives (with the same certainty) that they are consistent. Hence he has a mathematical insight not derivable form his axioms.
He distinguishes between "objective mathematics," i.e. mathematical statements that are true, and "subjective mathematics," i.e. statements that you can construct a proof for (and so can be proven to our subjective minds):
Evidently no well-defined system of correct axioms can comprise all of objective mathematics, since the proposition which states the consistency of the system is true, but not demonstrable in the system.
This is a statement of the second incompleteness theorem: If a system is consistent, there is no proof available in that system of its consistency. (Note that if a system isn't consistent, i.e. if some of its axioms contradict, then you can construct a proof of the consistency of the system, or of anything else, including that grass is blue and ice cream is God, because logically, anything follows from a contradiction.)
So if we know a system is consistent, then we know it without a proof. According to Turing, the mind is a machine that generates its knowledge through something going on under the surface that is much like the process of mathematical proof, so on this conception, we just couldn't know anything that we couldn't get through this kind of proof. The ultimate conclusion of Gödel's paper is a disjunction:
Either mathematics is incompletable..., that its evident axioms can never be comprised in a finite rule, that is to say, the human mind (even within the realm of pure mathematics) inifinitely surpasses the powers of any finite machine, or there exist absolutely unsolvable [mathematical problems].
He spends the rest of the paper discussing these two possibilities. Folks like Roger Penrose run with the first option here: we are not Turing machines, our minds are not computers in any sense.
Gödel's position in the paper is less than clear. Though he seems to disagree with Turing, most of the rest of the paper makes it sound like he affirms the latter option: mathematics is objective; it exceeds our grasp, as evidenced by the fact that there are mathematical truths that exceed our capacity to prove them (i.e. that "are not subjective," according to his weird definition of this term described above). This means that Gödel is a mathematical realist, math is not something we invented, but more or less something we discovered. Even though of course the symbols we use are our invention, once you lay out the system, you find it has all sorts of implications that you couldn't predict (i.e. that you can't prove in the system). The rest of the essay presents arguments against nominalism (the view that mathematical entities are just fictions in some sense).
We also read a less developed, more philosophical unpublished essay from 1961 called "The Modern Development of the Foundations of Mathematics in Light of Philosophy" (read it here with an introduction with helpful background from Dagfinn Føllesdal) that more directly takes on the mathematical realism vs. nominalism debate: there are "rightward" strains in philosophy like rationalism (in the sense of Plato, Descartes, and Leibniz) and theology, and "leftward" strains like empiricism. Despite the historical precedence of the leftward with the rise of science, he describes projects (like Mill's) that try to derive mathematical truths from experience to have been largely unpersuasive. Gödel sees the best option as a middle way between the two directions, and ends up recommending Husserl's phenomenology as a good option for exploring what this kind of intuition consists in, though he doesn't give us much detail in the paper about what advantages he thinks this will bring us.
You can buy Gödel's Collected Works, Vol. 3: Unpublished Essays and Lectures or read the two essays (with their introductions) online here and here.
For some additional information about what the incompleteness theorems do and don't imply, tryGödel's Theorem: An Incomplete Guide to Its Use and Abuse.