In addition to the other answers, this poem also makes use of apostrophe. Apostrophe is when the speaker of the poem addresses someone who is absent, an abstract idea (e.g., love, time), or an object (e.g., a vase, a flower). In beginning this poem with a question addressed to the speaker's lover, called only "thee" in line 1, Shakespeare employs apostrophe.
Others have identified the figure that occurs in line 5, with the "eye of heaven," but there is an addition to this figure in the next line. The speaker says, "often is his gold complexion dimmed" (line 6). Referring either to the sun, identified as the "eye," or to heaven itself, the speaker uses personification when he uses the possessive "his" and speaks of a "complexion."
Further, in the final line, the speaker refers to the poem, saying, "So long lives this, and this gives life to thee" (line 14). A poem, however, is not something that is alive—it does not "live"—but Shakespeare personifies it, saying that the poem will have a life. Moreover, even if the woman the speaker addresses dies, this poem's life will continue to keep her alive as well.
The poem begins with a comparison, so simile is the first figure of speech that you may want to identify and discuss. There's also "the eye of heaven" later in the poem, a fancy way to talk about the sun; you might identify and discuss this as periphrasis. Finally, "nor Death shall brag" presents death as a figure, so you can talk about personification here. I'm sure there are a number of other figures of speech at work in the poem (e.g. parallelism), and I hope that others will identify some of them.
In your analysis, I would encourage you to define briefly each figure of speech and make it clear where each figure of speech can be found in the poem; simply naming the figures of speech used in the poem may not be enough.
In addition to what the first answer identifies, I think there are at least a couple more figures of speech in this sonnet.
First, I would say there is metaphor. When Shakespeare talks about "thy eternal summer" he is not using the word summer in the literal sense. Rather, he is using summer as a metaphor for the peak of someone's life or, in this case, loveliness.
Second, I think the whole sonnet is an example of hyperbole. Surely she will not enjoy an eternal summer where she is always lovely. Surely the comparison to a summer day is exaggerated. I think he is overstating her charms to make his point.
In line 3, 'Rough winds shake the darling buds of May', an attribute of a living being is assigned to the 'winds'; hence is an example of Personification.
In line 5, 'the eye of heaven' refers to the sun. It is a case of Periphrasis, a round-about image; it can also be a Metaphor, and an instance of Personification.
Line 7, 'Every fair from fair sometimes declines' is an example Hyperbaton because of the transposal of the normal grammatical order.
In line 11, there is another case of Personification for death has been imagined as a braggart or empty boaster.
In line 12, we find a Metaphor of grafting: 'When in eternal lines to time thou growest'.
In the final couplet, there is a parallelism in the form of Anaphora, because of the repetition of the words, 'so long' at the beginning of two successive lines.