Let X be a complete metric space. Then the complement of a first category set in X is a set of second category in X.
What is explain in my class is "if the complement of a first category set is a set of first category, then the entire space would be countable unions of nowhere dense sets, which is not the case in complete metric space." Please explain it.
Firstly, $X$ must be non-empty for the claim as stated to be true. Instead of giving you the answer, here is a situation that if you understand it, and the definition of first and second category, then you'll understand your situation too.
Suppose that $S$ is an uncountable set. Then if $T$ is a countable subset of $S$, then $S\setminus T$ must be uncountable. Why? since the union of two countable sets is countable, so if both $T$ and $S\setminus T$ were countable, then $S$, their union, would be countable too.
So, back to your (non-empty!) complete metric space $X$. Do you know why $X$ is of second category? Can you show that the union of two first category spaces is of first category?