I should talk about this, since it often comes up in AI. Essentially, it is an operational definition of intelligence, which Turing put to answer the question ``can a machine think''? He introduced it in Mind, volume 54, Number 236 (1950). It's reprinted (abridged) in The Mind's I edited by Dennett and Hofstadter, (PSY AA:H 067) and in Computers and Thought edited by E A Feigenbaum and J Feldman (PSY: KH:F 032).
Do not confuse it with Turing machines. The two are separated by fifteen years, and have entirely different objectives.
Turing tried to answer the question ``Can machines think?''. He began by defining his terms. To judge the quality of something's thought, he said, we should be concerned with the symbols it emits; not with irrelevant properties like its physical appearence, strength, or chemical makeup. Our criterion will be how well these symbols compare with those that a human would emit in similar circumstances. As symbols are our concern, we should consider the most versatile symbol-manipulating machine we know: the digital computer.
So the question reduces to ``can we build a computer that is conversationally indistinguishable from a human?''.
To answer this, Turing invented the ``imitation game''. As often described, you have three rooms. There is an interrogator in room 1, a computer in either 2 or 3, and a human in the other room. Room 1 contains a teletype linked to the computer, and also to another teletype in the room where the human is. The interrogator does not know who is in 2 and who is in 3, only that there is a computer in one of them and a human in the other. He (or she) can type anything he likes into the link, to either room; and of course, he'll get a typed reply back. If, after ``conversing'' in any way he likes for as long as he likes, he can't tell which is the computer and which the human, the computer has passed the Turing Test.
The type of test Turing actually described is slightly more complex (see, e.g., Hofstadter). But it reduces to the same idea: a behavioural test for conversational indistinguishability from a person. Note again that the interrogator is allowed to type about anything at all (and in any language he wants). Not just arithmetic or chess, but poetry, politics, philosophy ...Why is this relevant to AI? In Penrose's view, we could distinguish an intelligent Turing-equivalent computer from a human by posing it a question which requires non-algorithmic methods (page 64).
In Dreyfus' view, we could distinguish an expert human from an intelligent computer (though not from an intelligent connectionist system) by looking at its expertise.
In Searle's view, we could not build a computer that's indistinguishable from a human. But the computer would have no subjective awareness. This is an odd view. See An Unfortunate Dualist by Smullyan in Mind's Eye.