Let $\sigma$ be a permutation of a set A. We shall say "$\sigma$ moves $a\in A$" if $\sigma(a)\neq a$. If $A$ is a finite set, how many elements are moved by a cycle $\sigma \in S_A$ of length $n$?
I am confused about the meaning of the question. A cycle can move $0-n$ element(s) in $A$. Is this correct?
Define a permutation $\sigma $ such that $\sigma(a)\neq a$ for $\forall a\in A$. Does the question mean how many this kind of permutation exist?
As discussed in the comments, the answer to your question is that a cycle of length $n$ "moves" exactly $n$ elements, i.e. exactly $n$ elements are not left fixed.
In general, we know we can factor a permutation $\sigma$ as a product of disjoint cycles, say of lengths $p_1,p_2,\ldots,p_k$. We can ask the same question: how many elements does $\sigma$ move? The answer is $$p_1+\cdots+p_k$$ It's as simple as that.