Eugene is your generic nerd character. He works on the party planning committee. He wears glasses with lenses thicker than the panes between prisoners and visitors at a maximum-security prison. And he's the target of bullying and homophobic jokes…which makes it really hard to feel fond of the T-Birds.
The moment that best represents why Eugene is bullied comes at the very end of the movie. Doody attempts to hit his teachers with a pie at the carnival. He throws and fails, and fails, and fails. Eugene gets it on the first try. Sorry, Doody, Eugene is legitimately better than you.
Doody retaliates by pie-ing Eugene. Not a good way to release your anger, dude. If he focused on his throwing arm instead of trying to humiliate someone in return, this guy could pitch in the major leagues. But hey, at least Eugene gets a free pie out of the deal.