Four-Eyes (as most call him) is the only Jewish boy in Sasha's class, wears glasses (hence where he gets his moniker), and his parents have recently been arrested. This makes him a triple threat, according to the children and Nina Petrovna, and he's picked on and made an outcast because of all three of these things.
When we first meet Four-Eyes, he's being pelted with tons of snowballs on the playground during a good old-fashioned game of firing squad. After a rallying cry of "Death to the enemy of the people!", the kids let loose in a flurry of frosty snow (12.2). Finkelstein doesn't fight back, though. He just lets it happen. He doesn't even stand up for himself when Sasha really whacks him a good one with a snowball, breaking his glasses and cutting his cheek (12.7).
Why's he so meek and mild? Probably because he cares about one thing and one thing only: the possibility of seeing his parents again. He pleads with Sasha to help get him into the prison (because Sasha's dad works there), but that's a no-go. Sasha's a little freaked out, since Four-Eyes mentions, "I'd do it for you if your dad were locked up" (14.19). This hits just a bit too close to home for Sasha (since his dad was just arrested the night before), but also teaches him about the utter brutality of the political system he's kowtowing to. It can viciously turn on even its most vocal supporters in a heartbeat, and leave even the good kids like Four-Eyes orphaned.
When Sasha tells Four-Eyes that his mom (Sasha's) wasn't a spy, that "[s]he was a real Communist" (14.16), Four-Eyes responds, "My mom and dad are real Communists, too [...] They are in Lubyanka prison now—enemies of the people" (14.17). You can almost hear Sasha's brain getting warmed up for a game of Follow The Logic. If even the best communists can get thrown into prison, it's no wonder Sasha's own father wound up in the clink. The sheer arbitrariness of it all helps push Sasha towards questioning the whole Soviet system.
By using a bit of civil disobedience and passive-aggressiveness, Four-Eyes finally gets closer to what he wants. He falsely confesses to breaking the nose off of the Stalin statue, because he thinks it will get him into Lubyanka to see his parents. Unfortunately, instead of getting the happy family cell that Sasha imagines, he'll soon find out his parents have been executed—just more victims of the rapacious Soviet injustice machine.