What does Angelou compare her hope to in "Still I Rise"?

In this poem, which demonstrates the speaker's inner perseverance through life's difficulties, Angelou creates varying metaphors to convey her strength. In one stanza, she writes,

Just like moons and like suns,With the certainty of tides,Just like hopes springing high,Still I'll rise.

Her hopes are like the moon, the sun, and tides. What do these things have in common? They are constants. The moon is always above us. The sun always rises. The tide always comes in. Hence, these are things that can be depended on, never questioned.
Like the moon, sun, and tide, the speaker knows that her hope will always soar high with a constancy resembling these natural images. Regardless of the lies that are told about her or the efforts of others to "trod [her] in the very dirt," she will bounce back with hope and rise above every difficult situation.
Choosing these particular objects also reinforces that it is only natural that her hopes remain through difficult situations, because each of these three references exist as forces within nature. Each has a powerful natural strength; the moon exerts its influence through tidal effects, the sun creates powerful energy and light, and tides are capable of moving significant objects with a powerful force.
Therefore, the speaker's hope embodies a power and a force that is capable of moving obstacles in her path. She will not be held down.

In stanza three of Angelou's powerful poem "Still I Rise," she compares her "hopes springing high" to the moon, the sun, and the tides. Essentially, through the use of simile, Angelou illustrates that her rising hopes are as reliable as the natural order of the earth. Then, in stanza nine, she claims that she is "the hope of the slave." This metaphorical statement refers to history rather than natural order but again demonstrates confidence in herself.
In a broader sense, although she only uses the term "hope" twice, the poem as a whole is a testament to Angelou's hopes and deep conviction that no matter what forces come against her, she will still have the strength to rise. As she returns again and again to this theme, she frequently uses both natural and historical references to support her ideas, which creates a sense of relatability with the reader.

Throughout the poem "Still I Rise," Maya Angelou compares her hope to several different things, each adding meaning to the poem. While the entire poem is about the hope of a black woman constantly attacked, Angelou uses two specific metaphors here. Before discussing the specific similes, however, it's important to discuss the reason why poets use figurative language. That reason is two-fold:
To provide a visual analogy for an intangible thing.
To bring out some emotion in the reader. 
In this poem, Angelou writes:

Just like moons and like suns,
With the certainty of tides,
Just like hopes springing high,
Still I rise.

This comparison of hope to the moon, sun, and tides provides this idea of the cyclical nature of hope. With every low, there will be a high. For every night, there will be a daytime. So, for every moment of hopelessness, there will be a moment of hope.
In the poem's last stanza, the speaker says she is "the dream and the hope of the slave." This metaphor is interesting because Angelou again provides major contrasts. Like she did earlier with the sun and the moon, she discusses the idea of the slaves and contrasts that with the idea of freedom that she will achieve for herself.
These two devices provide the poem with its inspirational tone. The speaker discusses the attacks she suffers and how she fights through each to end up rising above.

Angelou compares her hope to dust, air, the ocean -- anything that moves around, through, over an obstacle and is still essentially itself.  Her hope is rooted in the history of the African-Americans who were enslaved, beaten and killed, but whose population and tenacity remains strong in the America that tried to defeat it. The last two stanzas are a change in tone from the challenge and rhetorical questions of the majority of the poem: they state what she will DO, no matter what judgement confronts her, no matter what challenges lie in her path. The repetition of "I rise" intensifies the mood as we approach the end, and you can imagine Angelou herself, her voice booming forth with emphasis on each syllable.  It's not hard to see why this is one of the most common orally performed poems from contemporary literature -- a clear simile and an indelible image.