Figure of speech is a catch-all phrase that refers to the literary devices used in figurative language. Called tropes by some folks, figures of speech take meaning to another level (to the figurative, beyond the literal), and come in all shapes and sizes. Figures of speech include metaphors, similes, and personification.
In our everyday conversations, we often use "figure of speech" to refer to a common metaphor that everyone understands. So when you tell your mother that you love her to death and she looks horrified, you can say, "Aw, it's okay, Ma. That's just a figure of speech." But then again, she probably already knew that.