When you penalize someone, you punish them. A traffic cop will usually penalize a speeder with an expensive ticket.
A court might penalize a criminal with fines or time in prison, while a frustrated teacher is more likely to penalize the class clown with detention. Sports referees frequently penalize players by allowing the opposing team an advantage of some sort, like a penalty kick in soccer or a free throw in basketball. Penalize is rooted in penal, or "pertaining to punishment." The Greek root is poine, "penalty, punishment, or blood money."