Directors hold callbacks when they want to learn more about specific actors after an initial audition. The director may want to see how certain actors fit specific roles or just to see what else the actors can do. While callbacks can sometimes feel like an elimination round in a contest, it is important to keep in mind that the director could still cast anyone in any role, even if they have not been called back.
Often, directors only need to look at a handful of actors in greater depth for a few specific roles. Nobody is entitled to a callback, as they are purely a tool that the director uses to help them build on the decisions that have been made after the first round of auditions.
For a musical callback, actors are often asked to prepare sections of songs and dialogue (sides) for specific characters. Sometimes, a dance callback may be necessary as well.