What is a signature phrase in music?

It can be two things: First, it can be a short musical passage (a phrase) connected with a certain character or idea in the music. The operas of Richard Wagner are full of signature phrases (he called them "leitmotivs") identifying people, objects and places.

A "signature phrase" can also be a short musical passage that instantly identifies the work and it's composer - such as the first four notes of Beethoven's Fifth Symphony.