First-order logic uses only variables that range over individuals (elements of the domain of discourse); second-order logic has these variables as well as additional variables that range over sets of individuals. For example, the second-order sentence $\forall P\,\forall x (x \in P \lor x \notin P)$ says that for every set $P$ of individuals and every individual $x$, either $x$ is in $P$ or it is not (this is the principle of bivalence). ("Second-order logic", Wikipedia)
In the formal language of set theory, the axiom schema is: $\forall w_1,\ldots,w_n \, \forall A \, \exists B \, \forall x \, ( x \in B \Leftrightarrow [ x \in A \land \phi(x, w_1, \ldots, w_n, A) ] )$
Comparing and contrasting these two, I find that in the latter case, set $A$ can be considered as set of individuals that was quanitifed by universal quantifier. So, how is this different from second-order logic (in the first case, I would be referring to set $P$.)?