When people refer to “Goedel’s Theorem” (singular, not plural), they mean the incompleteness theorem that he proved and published in 1931. Kurt Goedel, the Austrian mathematician, actually proved quite a few other theorems, including a completeness theorem for first-order logic. But the incompleteness theorem is the one for which he is most famous.
To get some sense of the impact of Goedel’s Theorem on the mathematical community, consider how Herman Weyl, perhaps the greatest mathematician of the first half of the twentieth century, reacted to it. According to Weyl, up until Goedel’s Theorem, mathematicians were optimistic that all questions in mathematics could be definitively answered either one way or another (see his contribution to Oldenbourg’s Handbook of Philosophy).