Sometimes we use the phrase 'figurative language' to mean 'figure of speech'. Sometimes we use it to mean 'language that is not literal'. The two do not precisely overlap, because there are figures of speech that are not figurative in the latter sense. Some examples:
Allusion: 'Allusion' covers a lot of things, but at least some of them are not figurative. One example is what we can allusion of citation. Suppose I were to say, "He was reading of the Cave and the Divided Line." This is an allusion to Plato's Republic, of course, although it is not mentioned as such. Much of our literal discourse depends for its intelligibility on our ability to understand these allusions to background information.
Grammatical Figures: Many figures of speech are merely unusual grammatical structures. For instance,
He drove off in a high dudgeon and a Cadillac
This is usually called a zeugma; it's a purely structural way of using prepositions in an odd way, for effect (e.g., humor). This is called tmesis: "What man soever" (we've put one word between the syllables of another word).
Enargia: Some figures of speech are conventional ways of structuring some kinds of information in order to make the description especially vivid. For instance, a catalog, like Milton's "Rocks, caves, lakes, fens, bogs, dens, and shades of death".
Figures of Repetition: Some figures of speech are just repetitions of sounds. Alliteration, for instance. You can alliterate in literal language. Other figures of speech are repetitions of words. For instance, here's an anadiplosis from Shakespeare's Richard II:
The love of wicked men converts to fear,
That fear to hate, and hate turns one or both
To worthy danger and deserved death.
In an anadiplosis one repeats a word near the end of one phrase or clause at the beginning of the next. Nothing figurative (in the second sense) about that.
Or from Hamlet, here's an epizeuxis:
Words, words, words...
Epizeuxis is just straight repetition of a word for effect.
Figures of Reasoning: Some figures of speech have to do with the organization of an argument. For instance, the figure of anthypophora is the posing of questions that one then immediately answers, e.g.,
He has been rewarded. And what is his reward? It is the satisfaction of a job well done.
Pysma is the figure in which one successively asks a number of questions. For instance:
What is pseudoscience? How can we distinguish it? Can we be rid of it, and how?
Apophasis is a figure of reasoning in which one successively rejects a number of alternatives. For instance:
You cannot affirm that the government has all political power. You cannot affirm that the people have all of this power. Nor can you affirm that the power is invested in some third thing. Thus you must admit that the power is shared.
The list of examples could be extended considerably -- the figures of speech that are not figurative are legion.
