Notes from the Underground is one of the earlier examples of realist literature. Rather than focusing on, well, "the beautiful and sublime," Dostoevsky paints a gritty portrait of a shabby man in a dirty hole in the ground. He's not trying to rise above the grisly details of dirty reality – he's putting it in our face.
But enough about realism. Philosopher and scholar Walter Kauffman called Notes from the Underground "the best overture for Existentialism ever written." So while the late-19th century German philosopher Friedrich Nietzsche wasn't really doing his thing yet, and the mid-20th century French existentialist Jean-Paul Sartre was a mere glint in his father's father's eye, we can see how they might have picked up a thing or two from Fyodor Dostoevsky.
Dostoevsky wasn't strictly speaking a real honest-to-goodness official "existentialist." In fact, that term didn't exist yet. And in fact, once it did exist, most people called "existentialists" adamantly denied the label.
But the point is, we can identify many existentialist ideas tenets in Notes from the Underground. And if you want to know what these tenets are, check out Shmoop's Themes.