This story is perhaps most significant for its introduction of a character type: the Underground Man. The Underground Man resurfaces in short stories, plays, novels, and later movies, again and again, and if you watch for him, you begin to see him everywhere there's an underdog who doesn't succeed in the end. His frustration with the events in his life are taken as a critique of the utopian ideal. Instead of a rosy view of human nature, Dostoevsky's Underground Man sees people as "irrational, uncontrollable, and uncooperative," and the reader gets the sense that he may be on to something.
There's an audio archive of the story at http://librivox.org/notes-from-the-underground-by-fyodor-dostoyevsky/ - sometimes it's nice to hear these things read. There's a wonderfully comprehensive study guide, including a section about how this short story was received at the time, at http://www.middlebury.edu/~beyer/courses/previous/ru351/novels/UGMan/ugman.html.