What is common matches between first order logic and the second order logic? in other word what in first order logic is in second order logic?
Every sentence in first-order predicate logic is also a sentence in second-order predicate logic. (Assuming the langues are compatible, i.e. you have the same predicate and function symbols, of course).
And if it's provable in first-order predicate logic, it's provable in second-order predicate logic. (Now assuming the languages are compatible, and the axioms you use in second-order predicate logic are a superset of the one in first-order predicate logic, of course).
The converse is not true, i.e. second-order predicate logic is (way!) more powerfull than first-order predicate logic. You can, for example, uniquely define the integers in second-order predicate logic, but you cannot do so in first-order predicate logic.