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Turing Machines
The Turing machine can compute anything that can be computed. It is the very definition of computation and the fundamental tool for reasoning about computers. You really need to know what it is all about.
A Turing machine is an abstract computational model that performs computations by reading and writing to an infinite tape. Turing machines provide a powerful computational model for solving problems in computer science and testing the limits of computation — are there problems that we simply cannot solve.
An implementation of a Turing machine (Above)
Additional Reading
www.i-programmer.info/babbages-bag/23-turing-machines.html
Of course, Wikipedia's own article on this topic is always worth a read.
Check out the following site as well which lets you trial an online/virtual Turing machine! Pretty cool: http://turingmachine.io/