In literature, allusion is a subtle reference to an object or event, which depends on the sophisticated reader to recognize it. The densest example is James Joyce's Finnegan's Wake, about which entire volumes have been written to explain its historic and literary references. Poets like Virgil used allusion to pack a lot of power into a few words.
More familiar today would be this technique as used in films. Directors may slyly allude to movies they admire by creating dialogue or scenes that imitate them in a way that fans will recognize (known as homage). In these cases, allusion provides a sense of depth and context, and the new work becomes associated with the emotion and ideas of the original.
According to Jim Canterucci, author of Personal Brilliance: Managing the Everyday Habits that Create a Lifetime of Success, only 35 percent of Americans have read a book since high school. And 20 percent say they have never been in a bookstore. This creates a barrier to the use of allusions to literature.
For speakers, the purpose would need to be simpler and more obvious to a wider group, providing resonance with a well-known aspect of culture. The Reverend Martin Luther King Jr. began his “I Have a Dream” speech with, “Five score years ago,” an obvious allusion to President Lincoln's Gettysburg Address opening, “Four score and seven years ago …” The purpose was to suggest a parallel cross-roads for the nation in these two struggles, the Civil War and the movement for civil rights for African Americans.
With Americans reading fewer books, magazines, and newspapers and knowing less and less about history and literature, allusion has challenging limits today. But most everyone will know the basic idea behind references to Pandora's box or “between a rock and a hard place,” even though these allude to Greek mythology. Certain expressions from pop culture are so widely known (such as the Wicked Witch of the West or Catch-22) that they can be safely used, often tongue-in-cheek.
If you have time to let your speech gestate, your subconscious may bring up such cross-references and listeners who understand them will appreciate your presentation on another level.