Metaphor: an indirect comparison between two or more seemingly unrelated subjects that typically uses "is a" to join the first subjects
Symbolism: using something specific to stand for something else, especially an idea. So it's representation instead of comparison.
Allegory: a figurative mode of representation conveying a meaning other than the literal. In other words, it's all about the figurative rather than the literal connotations of a phrase.
Allusion: a stylistic device in which one refers covertly or indirectly to an object or circumstance that has occurred or existed in an external context. So in other words, a reference to something outside the scope of the story being told.