There are several different ways to talk about the genre of "Richard Cory," depending on which aspects of the poem you wish to emphasize.
First, it is a "lyric" rather than "epic" poem. This distinction is an ancient one. Epics were typically defined as long heroic narratives, and lyric poetry was a shorter and more personal genre, dealing with individual experiences rather than the great events of history.
Second, it can be described as a "narrative" poem, meaning that it tells a story rather than simply expressing an emotion or describing something.
Although it is not a traditional oral ballad, its structure—in particular the use of open quatrains, simple narrative, and everyday experience—owes something to the ballad genre.
Finally, it is a "realistic" rather than romantic poem, talking about the everyday life in a small town.