Turing's claim
Turing held that computers would in time be programmed to acquire abilities
rivalling human intelligence.
As part of his argument Turing put forward the idea of an 'imitation
game', in which a human being and a computer would be interrogated under
conditions where the interrogator would not know which was which, the communication
being entirely by textual messages. Turing argued that if the interrogator
could not distinguish them by questioning, then it would be unreasonable
not to call the computer intelligent. 
Turing's 'imitation game' is now usually called 'the Turing test' for intelligence.
Turing's 1950 paper has given rise to a large literature, surveyed by:
Chalmers' bibliography.
Robert E. Horn at Stanford University who has produced large MAPS of the arguments giving a 'visual language' for discussing whether computers could think. These maps form part of a larger project for mapping arguments.
This page of the American Association for Artificial Intelligence website discussing the Turing Test in the course of an extensive exposition of AI topics.
War Games and Mind Games
Alan Turing was based at Bletchley Park, Buckinghamshire, while acting as the leading cryptanalyst of German ciphers during the Second World War. It was during this period that he formulated the ideas that emerged after the war as Intelligent Machinery, and would now be called Artificial Intelligence. He must have been influenced by the astonishing power of mechanical methods at Bletchley Park.
The codebreaking work at Bletchley Park was highly secret and discussed only with those directly involved; but Turing used game-playing, particularly chess-playing, as a close analogy.
Results
Turing claimed that in 50 years computers would "think", just as humans do. However there have been many experiments over the years that prove his theory was unjustified and wrong. Computers are programmed to do what they are told and nothing more (well , most of the time!)
