Layman (level 1)
Hypothetical simplistic mechanical processor. Despite its simplicity, a Turing machine can theoretically perform any algorithm a real computer can.
Beginner (level 2)
Mathematical model of computation that defines an abstract machine. Consists of an endless strip of tape, a reading device and a table of instructions. Mathematically proven to be capable of any computation.
A programming language is said to be Turing complete if it's as powerful as a Turing machine.
Expert (level 3)
Turing machines are an abstract model of computation. They provide a precise, formal definition of what it means for a function to be computable.
A Turing machine can be thought of as a finite state machine sitting on an infinitely long tape containing symbols from some finite alphabet. Based on the symbol it’s currently reading, and its current state, the Turing machine writes a new symbol in that location (possibly the same as the previous one), moves left or right or stays in place, and enters a new state. It may also decide to halt and, optionally, to output “yes” or “no” upon halting. (source: Introduction to Algorithms, Notes on Turing Machines, Cornell University CS 4820)
A Turing machine is a pushdown automaton without the last-in-first-out (LIFO) requirement of its stack.