Full Idea
First-order logic is distinguished by generalizations (quantification) only over objects: second-order logic admits generalizations or quantification over properties or kinds of objects, and over relations between them, and functions defined over them.
Gist of Idea
First-order logic concerns objects; second-order adds properties, kinds, relations and functions
Source
Michael Dummett (The Philosophy of Mathematics [1998], 3.1)
Book Reference
'Philosophy 2: further through the subject', ed/tr. Grayling,A.C. [OUP 1998], p.134
A Reaction
Second-order logic was introduced by Frege, but is (interestingly) rejected by Quine, because of the ontological commitments involved. I remain unconvinced that quantification entails ontological commitment, so I'm happy.