According to Wikipedia second order logic allows quantification over sets of individuals and thus goes beyond first-order logic, e.g. in expressive power.
On the other hand some sort of quantification over formulas is allowed even in first-order theories. Formulas correspond to definable sets, and quantification over formulas (= definable sets) is typically realized by some axiom schema.
In the context of set theory quantification over sets of sets seems to be no problem, because sets of sets are themselves sets and thus belong to the domain of discourse. (Is this correctly stated?) Quantification over arbitrary collections (classes) of sets instead goes beyond first-order set theory. (Does this mean, that the Wikipedia definition is too narrow or even flawed?) But again the result of quantification over definable classes seems to resemble first-order logic.
In another Wikipedia entry one reads, "that quantification over all first order formulas cannot be formalized in the language of [first-order] set theory". A specific example of this fact is, that the notion of "definable sets" (involving an existential quantifier over formulas) is not first-order definable.
Why can some quantifications over formulas be formalized/realized in the language of a first-order theory (e.g. by axiom schemata), and others can not?