Realism with a big "R" can mean rather different things when applied to different authors. At its most general, though, it's a genre that tries to depict unromanticized ordinary life, the real life of regular, unremarkable people. The characters can't be rich and famous, or startlingly original and breathtakingly ingenious, and they can't be heroes. If they were any of these things then they wouldn't be ordinary. Amazingly exciting or unbelievable things can't happen to them. And they also can't be the kind of wonderfully wholesome characters that make it seem like just plain ordinary life holds the keys to perfect happiness. That would be romanticized ordinary life.
The Death of Ivan Ilych is definitely a Realist work. This is a story about a middle-class official with an unhappy marriage and shallow friends whose greatest joy is bridge games. Ivan dies because of a fall he took while setting up curtains. That's unromanticized ordinary life right there. Tolstoy's particular brand of realism is also characterized by psychological realism. He tries to describe the thoughts and emotions of his characters as directly as possible. For this reason, Tolstoy usually has the effect of coming across as a master shrink who understands his characters in ways they themselves could never do. Because Tolstoy himself was such a good psychologist (supposedly, he used his diary as a means to dissect his own mind), he's able to create believable and ordinary characters.
The novella is also a satire through and through. Tolstoy wants us to laugh at, and despise, the reality he depicts. (For more on how that reality reflects his own historical circumstances, see "Setting.") The characters seem to be so petty and false that at times it's humorous to the point of being unbelievable. Is this at odds with Tolstoy's realism? Tolstoy would probably say no. In the real world people are that petty and laughable, and die for pointless things all the time. It's only our illusions about ourselves – just like the illusions Ivan and all of his friends have about themselves – that make his satirical world seem exaggerated.