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Other answers are right but I think they are missing an important point that most people, even some experts on Alan Turing, tend to somehow overlook (sometimes unfortunately in favour of Turing’s personal life rather than his work).
And I am not talking only about Turing’s key role leading the effort to decipher the Enigma code during WWII. Not only his pioneering ideas on morphogenesis, a process by which many complex systems evolve producing a transient of sustained highly organized complexity producing patterns by the mechanism of reaction and diffusion. Not even his cutting-edge ideas on machine intelligence and his imitation game as a test, having impact both in the Philosophy of Mind and in some flavours of modern AI. But because of his insight into the concept of computation, both in the negative in the form of establishing the limits of computation (and thereby the limits of mechanical knowledge itself), and in the positive leading to the full understanding of the concept of computation universality. The negative results was equivalent and only next (in time) to Gödel’s work, and the positive result (that computation universality) is equivalent to Church’s, Post’s and others, but not next to anyone else, and not in e.g. Gödel’s approach (at least not directly).
While Alonzo Church’s, Emil Post’s and later others came up with equivalent models of computation, it was not but Alan Turing that fully grasped it to the point of explaining it in full detail and constructed (in paper) one. He found that there is a very general form of computation that can carry out and simulate any other computation of the same very general type (basically what we would characterize today as mechanical and digital computation). This would be similar to, for example, to crediting Church with the first formal functional programming language in the form of his lambda-calculus, which we rightly should.
To illustrate this let me explain Konrad Zuse’s achievement and involvement. Zuse’s was certainly among the first, if not the first to actually build a potentially universal mechanical computer. However, Zuse was not really aware that his machine was universal nor he was trying to build one. In fact, his computer turned out to be universal and was proven much later to be so by a Mexican-German researcher, Prof. Raul Rojas, not without having to use some minor technical tricks to prove so of which Zuse would have been completely unaware of.
Some others, including Gottfried Leibniz and Ada Lovelace, may have—and I personally believe they did—fully grasped the possibilities of computation universality but neither they were able to prove it really possible nor did they formalize it.
So, in summary, Alan Turing can safely be credited with being the actual first person that grasped the width and breadth of one of the most important concepts in history, that of computation universality and who was first able to formalize it, giving birth to what we formally know today as Theoretical Computer Science.
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