Well, his invention of the abstract concept of a so-called Turing machine (like the one in the Google doodle) threw light upon the limitations of algorithmic devices such as computers. More precisely, to use a horrible German term, Turing demonstrated that the Entscheidungsproblem cannot be solved. And what is this exotic beast? You might call it the "mission accomplished" problem. Like George W Bush with his war games, computers will remain forever incapable of determining beforehand whether or not a certain computing challenge can indeed be handled successfully. Turing taught us that the only way of knowing whether or not a computer can handle such-and-such a complex challenge is to set the machine into action and see whether or not it soon halts with a solution.
You might say that Turing proved that the proof of the computer pudding is in the computing.