Let's get down to brass tacks. An allusion is, plain and simple, a reference. You'll find allusions (or shout-outs, as we like to call them) when the book you're reading makes a reference to something outside of itself, whether another work of literature, something from pop culture, a song, myth, history, or even the visual arts.
Why use allusions? Because they connect literature to other pieces of literature (or art or music or history or whatever). Allusions deepen and enrich a work's meaning, and are a form of intertextuality, so they help books talk to each other.
Examples? William Shakespeare is the king of being alluded to and referenced in literature. The title of William Faulkner's The Sound and the Fury is an allusion to a line from Macbeth, and when we read Faulkner, if we keep Macbeth in mind, Faulkner's meaning just might be enhanced.