As mentioned above, you really must read Notes from the Underground/Notes from Underground to experience the book as gestalt. If you keep your eyes open, you'll notice tons of carefully-veiled insights. The book is divided into two main parts: the manifesto (only about 50 pages in my edition) and the story (the other 100). Anyhow, onto the meat of the book.
Dostoyevsky's psychology and philosophy are quite simply stunning in this book, and because of this it is quite difficult to pin down exactly what Underground man is trying to say. But he says quite a bit in Part I, "Underground." He does some wonderful commentary on "advantages," namely that the most important one is, in essence, spite; he almost worships human irrationality. Human consciousness is a disease, but all men love their diseases. Their moans and screams give him control of those around him, allowing him to bring them to his agony and pain. He also comments on the state of modern man, questioning as to whether or not we're (19th century Russia) really out of the barbarian age. He questions vengeance and justice. He violently assaults humankind's view of itself.
Now, Part II of the book, "On the Occasion of Wet Snow," is a story of his youth, beginning at age 24. Dostoyevsky puts the manifesto of Underground man into practice as we learn why our anti-hero is what he is. The vengeance that he has for a certain officer is simply spell-binding; page after page goes on to describe how Underground man plots his revenge on the officer for moving him out of the way at a tavern. This part of the story mostly deals with his concept of perverse pleasure. The second part deals with a classmate reunion and his night with a whore. We learn what Underground man meant when he talked about his attempts to leave his underground. As the story progresses, he spends a night with Liza. I can't describe it very well. You just have to read it. The profound--and often contradictory--nature of Underground man truly shines in this final portion, as he even makes pokes at determinism and the psychology of personality. Although it is easier to read and understand, this second portion of the book is by no means any less deep as the first.
As I mentioned above, there are many little interesting insights hidden throughout the book, and perhaps my favorite one is where he comments on Cleopatra enjoying poking her slave girls' breasts with golden pins. Although 19th century Russia is more "civilized" than ancient Egypt, he notices that they do much the same thing (physicians with syringes). Underground man feels that our civilization has in no way made our barbarism any less.