Definition
A Turing machine is an abstract model of computation consisting of an unbounded tape divided into cells, a head that can read and write a symbol on one cell at a time, and a finite table of rules. It is a mathematical idealisation, not a physical device.
At each step the machine reads the symbol under the head and consults its rule table, which tells it what symbol to write, whether to move the head one cell left or right, and which internal state to enter next. Despite this minimal toolkit, the machine captures everything we mean by mechanical computation.
Why it matters
How it works
To run a Turing machine you write the input on the tape, set the head and the internal state to their start positions, and then repeatedly apply the rule that matches the current symbol and state. The machine halts when it reaches a configuration for which no rule applies, and whatever then sits on the tape is its output.
Priest uses the Turing machine as the bridge between informal talk of algorithms and rigorous theorems about their limits. Because the model is fully specified, one can prove that certain problems — the halting problem chief among them — cannot be solved by any Turing machine, and therefore, granting the Church-Turing thesis, by any algorithm whatsoever.