I may or may not have read this as an undergrad, but I didn't remember a thing about it if I did. This in no way indicates the success of the story but rather my awful memory.
Of the stories in the collection (so far), this one is the saddest, although it still has shades of humor. It's one of those stories where you see something awful coming and fervently hope against it then feel crushed when it happens anyway.
As with other stories, "The Overcoat" features a member of the government service. He's of low rank but doesn't even want to rise in rank when it's possible. He just loves copying (his job). He doesn't make much, and he leads a solitary life. Others make fun of him, especially his threadbare coat that doesn't keep him warm when winter comes. He scrapes money together, essentially starving himself, until he has enough to pay a tailor to make a new coat. Finally it's finished, and he's ecstatic about his new coat. His colleagues even throw a party. You just know something terrible is going to happen to this coat.
Going home after the party, his new coat is stolen, and the police officer is useless. Everyone he tries to get help from is useless, and one Important Person he pleads his case with rants at him so badly that he gets sick and dies! His ghost starts haunting the streets, stealing coats from the living. The Important Person who was a dick to him feels badly about how he acted. One night when the Important Person is going to see his mistress, the ghost steals HIS coat, and the ghost is never seen again.
I'm not sure I understood the end of the story--there's a different ghost haunting the area now, but I'm not sure if he's supposed to be one of the other characters?
Regardless, the story is poignant and off-kilter like Gogol's other works, and I definitely enjoyed it.