In order to understand the meaning of your essay question, you have to first understand the allusion to "Richard Cory" and the meaning of "allusion."
An allusion is a literary device by which a large concept (like the characteristics of a well known person) are brought to mind through the introduction of a simple word, phrase, or expression. In the case of your essay question, the allusion--the single word, phrase, or expression that is meant to call to mind a larger concept--is "Richard Cory." A readily understandable example of allusion is the word "Superman." This word calls to mind the larger concept of an individual who is unique in the culture and who is dedicated to bringing good to humankind by braving dangers and triumphing over evil so that peace, good and harmony may be given to an individual, a group, or the whole world (using the allusion "Superman" delivers the above concept to mind with much less effort than it takes to explain it).
Who is Richard Cory? Richard Cory is the protagonist of a narrative poem by Edwin Arlington Robinson (I'm supposing this poem was part of your class syllabus), and the poem is titled after him, "Richard Cory." In this poem, we encounter a bitter personal irony that shocks and dismays both the narrator of the poem and the reader.
Cory is established in the lines of the poem as the picture of perfect refined and elegant manhood: he is the man every man wants to be and the man every woman wants to meet (and marry). He is refined, gentlemanly, admired, though never pushy, brash, arrogant or superior to those he spoke with: "And he was always human when he talked;...." While he "fluttered pulses" of the women, he served as the role model for the men as he was refined, favored, rich, "imperially slim":
So on we worked, and waited for the light,
And went without the meat, and cursed the bread;
Then the narrator learned--and the reader learns--that Richard Cory had a different assessment of himself (perhaps he wished his admirers were more "human" in their approach to him) because one night, while the women had fluttered pulses and the men tried to be like him, Richard Cory went home and put an end to himself.
Your task is to think of someone in contemporary society who might be comparable to Richard Cory: Who is someone that all admire and look up to, women and men alike, and who all wish they could be like, but who, harboring secret disappointments, failures, bitterness, put an end to their own life (in today's society, it might be a woman as well as a man, I suppose). One example comes to mind: Vince Forster, friend and adviser to Bill Clinton during the Clinton Administration. To the shock and dismay of all, this notable man, who seemed to be living the hyperbolic "Camelot" life as royal liege, wrote a letter of resignation, tore it to bits, then ended his life.