An article by A.M. Turing
published in 1936
in the Proceedings of the London Mathematical Society
, ser. 2. vol. 42 (1936-7), pp.230-265
(available on the web on abelard.org)
This article introduces the Turing machine, and uses it to answer Hilbert's Entscheidungsproblem in the negative. This relies on the idea that anything computable at all is computable by a Turing machine, a statement now known as the Church-Turing thesis.
Detailed explanations of its subject matter can be found by following the softlinks below.
The article is exceptionally easy to read: great pains are taken to avoid dense or technical language.
In its formulation, it foreshadows Turing's later article, Computing Machinery and Intelligence (Mind, 1951) on whether a machine can have a mind.