Figures of speech are a great many different uses of language -- allegories and similes, hyperbole and metonymy, and alliteration and onomatopeia are all included under this heading (as are many others.)
But the most dominant figure of speech in "The Sunne Rising" is personification. Immediately the poet addresses the sun as a human being -- and he takes issue with the sun's behavior (he is angry that the sun has risen, because it ends his time in bed with his beloved)!
Busie old fool, unruly Sunne
Why dost thou thus,
Through windowes, and through curtaines call on us?
Must to thy motions lovers seasons run? (1-4)
This trope (another name for certain kinds of figures of speech) continues through the entire poem. This is also an example of apostrophe (the same word as for the ' mark used for contractions and possessions, but in this case the word is used to describe a kind of poetry in which the poet addresses inanimate objects -- such as here, the sun). The personification of the sun is shown when the poet gives the Sun human qualities (calling it, for example, "busy" and a "fool", or that the Sun can "Goe tell Court-huntsmen, that the King will ride" (7)); the apostrophe is that the poet addresses, or pretends to speak to, the Sun at all. So, in this poem, Donne is using both personification and apostrophe. The poet uses other figures of speech in this poem, as we will see, but the personification and the apostrophe are the dominant ones, and the rebuke of the sun is the idea which contains Donne's claims about the glories of his beloved and himself.
Donne uses hyperbole, also -- a form of poetic exaggeration.
Thy beames, so reverend, and strong
Why shouldst thou think?
I could eclipse and cloud them with a winke,
But that I would not lose her sight so long; (11-14)
Of course, Donne, no matter what he says, cannot "eclipse" the beams of the sun. He is saying that the brightness and the beauty of the eyes of his beloved, and the importance and grandness of their love makes sunlight dim in comparison. It's a broad, daring, and expansive hyperbole, and it implies not only the poet's strength of love-ardour, but also, perhaps, his foolhardy youthfulness. These superlatives (another kind of figurative language) continue all the way to the end of the poem -- his beloved, and his love for her, are the greatest and most important things on earth -- in fact, in the end, the poet says that they are the whole earth. A superlative, indeed!
These superlative ideas are all a kind of metaphor, extending and increasing in importance from the beginning to the end of the poem (a form of climax) -- the end, in which all of existence is contained within the bed of the two lovers. The ruling trope of the personification of the sun is brought to a close near the end of the poem, when the poet directs the sun to only warm the lovers, because they are, contained in themselves, the entire world. "...and since thy duties be/To warme the world, that's done in warming us." (27-28) It is bold, audacious writing, typical of Donne, and these are only some of the figures of speech he employs in this famous poem.