I would like to show that a problem $A$ is $NP\text{-complete}$. So, I am trying to reduce 3-SAT problem to $A$. Reduction is kind of function, let say $f$. What is necessary complexity of $f$? Probably, it has be in $NP$. But, does it mean that $f$ must be a non-deterministic Turing machine working in polynomial time? 
Reduction problem to another problem
complexity theory;np complete;reductions
No, $f$ need not be a NDTM.  To prove a problem is NP-complete by reducing another NP-complete problem to it requires only a Karp reduction, which is a polynomial-time deterministic reduction.  So $f$ would live in the complexity class FP.