Approving describes a look, a voice, a nod, or a comment that shows you are on someone's side: you agree, you support, you think they did well. An approving smile says yes, good without a speech. It is the -ing form of approve, used here as an adjective. The opposite is disapproving — a frown, a silence, a tight mouth. Though actually, an approving look is quieter than the noun approval; it is often a face in the moment, not a stamp on a form. People use it about reactions, not about official permission.