You can think of it as a theoretical machine introduced by Alan Turing which can compute anything that can be computed. If something cannot be computed by Turing machine, then it means that it cannot be computed at all. An important aspect is that it is defined as a simple machine with infinte length tape and a moving head, which can move in any direction. Tape is made up of cells where inputs/outputs are stored. There is also another related proof that says problems are uncountable and programs are countable which means there exist some problems which cannot be solved at all. I would recommend the book Introduction to Automata Theory, Languages, and Computation by Ullman et al. It explains from very simple machine called finite automata to Turing machine. For a beginners introduction Wikipedia entry is still good