A non-deterministic turing machine can interprete multiple instructions (write 5 to tape and go to state 2 OR write 3 to tape and go to state 19) and (depending on interpretation) use the instruction that gives the result fastest or explore all instruction branches at once.
Note that the "OR" in there is not a classical if-else, it means literally the machine can pick one. There is no memory value that tells it which is correct.
Basically, problems that are polynomial on a NDTM will likely be NP on a normal machine (with some exceptions depending on the problem).
Re2: As above, verifying is actually easy as you can then simply follow the execution branch the NDTM picked on a normal deterministic turing machine.
Atleast this above is what I learned in CS course 2nd semester.
