Non-determinism like in PDA (partially NFA) for one input configuration has several possible outputs.
The non-deterministic TM is like TM but with a finite number of choices of moves; may have more than 1 move with the same input “current state & current symbol”.
The non-deterministic TM accepts the input w if there is at least one computation that halts normally for the input w.
Non-determinism is more powerful than determinism for pushdown automata. But it makes no difference for finite automata.
Quite surprisingly, the deterministic and non-deterministic Turing machines are the same in power.
If a nondeterministic Turing machine accepts a language L, then there is a deterministic Turing machine that also accepts L.
There are many other similar machines, which might look more or less powerful, but which can be seen to be equivalent in power to the simple Turing machine (recognise the same recursive enumerable languages generated by the unrestricted grammars) − adding more tapes, more control units, . . .
Equivalence is shown by describing how a simple one tape Turing machine can be used to emulate them, and vice versa.