It helps to know some historical background in order to understand who the poem is addressed to. Pushkin was a Russian writer who was interested in famous Russian leaders such as Peter the Great. The title of the poem is taken from a statue of Peter the Great on a horse.
The poem is about the attempt to reconcile building a vast and powerful Russian empire while addressing the needs of the individual citizen. Scholars debate the success of this reconciliation. Some claim that it shows how the Russian empire puts the empire first and the individual second. This analysis is mostly based on the fact that Evgenii (also spelled Yevgeny) symbolically dies at the hands of Peter the Great in the conclusion of the poem. (Evgenii is often translated as Eugene.)
But however one interprets the poem's statement about empire, it is certainly addressed to Russian (and later, Soviet) citizens. Some say it is a national epic. That is, it celebrates Peter's vision for a Russian empire and city (St. Petersburg) but that it also showcases the plight of the individual as the empire develops.
In the first two stanzas of the introduction of the poem, Peter stands on the banks of the Neva River, contemplating the construction of a new Russian city: St. Petersburg. Thus, the poem's speaker is thinking back to how Peter envisioned his city. This city would be constructed from land the Russians took from Sweden. At the time, it is uninhabited, a land of "gloomy wood and swamp." Like the building of the city, the Russian empire would rise up from nothing into something glorious. The rest of the introduction is an ode to Peter.
Part 1 is about Eugene (Evgenii). He is a poor, young man in the city. He is in love with a woman named Parasha. He goes to sleep and wakes up to find the city flooded.
In Part 2, Eugene uses a boatman to seek Parasha's home. He finds it completely destroyed and is devastated. He turns his anger on the statue of Peter the Great. Then in a moment of magical realism, the statue becomes angry and pursues him. The poem ends with the image of Eugene's dead body. This illustrates the dichotomy of praising Peter the Great but also recognizing how the individual is sacrificed for the sake of the empire. Note that the flood is actually based upon a flood that occurred in 1824. However, the flood is also symbolic here. Eugene loses everything in the flood and he looks at the statue as an emblem of the empire and feels that the empire is oblivious to his life and concerns. He and other individuals like him are swept away by the flood of the growing empire.