The most important figure of speech in this poem is the simile, which compares two different things using the words "like" or "as." In the first stanza, the speaker compares his love ("Luve") to a "red, red rose, / That's newly sprung in June." That is to say that his love is like the rose at its most vibrant state in summer, its "reddest" color (this is why the word "red" is repeated). As the rose is at its reddest color, his love is at its fullest feeling. The speaker uses another simile to compare his love to a melody that's played in tune.
In the second stanza, the speaker uses a simile again. He notes that his love is as deep as his lass is beautiful ("bonie"). If she is very beautiful, his love is very deep.
In the second and third stanzas, the speaker uses hyperbole (exaggeration) to say that he will love her until the seas go dry and until "rocks melt wi' the sun." Although this seems like an eternity, it does have a limitation: "until" the seas dry and rocks melt with the sun.
O I will love thee still, my Dear,
While the sands o' life shall run.
These lines are ambiguous. He could be saying that he will love her beyond the end of time, which is another example of hyperbole. He might suppose he could love her beyond the end of time (love being beyond time) but the best way he can express this is with this exaggeration. These lines could also mean that he will love her until the end of time; still, an example of hyperbole since the end of time seems so far away.
In the last stanza, after professing his love, the speaker notes that he is leaving for a "while." He promises to return even if it were ten thousand miles. Quite difficult to travel ten thousand miles in 1794; therefore, this is also hyperbole. Burns uses these exaggerations (hyperbole) and similes to show the depth and degree of love.