Finally Turing machine is introduced. With NFA\PDA in mind, readers should be able to understand Turing machine without too much effort. Again, Prof. Michael Sisper deserves a title of Teaching Guru.
Turing machine has a magic 'Tape' which acts like a RAM, an infinite RAM - R\W, moving forwards\backwards. It is much more powerful than stacks used by PDA and here comes the capacity of computation. A typical TM utilize only one tape, and a multi-tape TM has the same capacity comparing with a one-Tape only TM. Why? Just merge those tapes into one tape and modify your transition function. Non-deterministic TM: it reproduces itself by the number of possibilities and explores each branch until an ACCEPT is encountered. And so far, except Quantum Computer, all other computational models can be reducted to a TM.
I never knew there's lower level than an algorithm, well, a Turing machine is. By encoding data structure used by an algorithm, TM can run that codes - all right, Shannon's information theory here..