When Britta and Annie square off in their cheerleader outfits, the Spanish vocabulary words on the chalkboard behind them ("pelear, pecado, pecho, pechuga, peligro") translate as "to fight, sin, chest, breast, danger." The word above them, "strazio," is Italian and means "torture" or "torment."
Abed mentions how Hawkeye of M*A*S*H kept his humor and charm "even in the 11th year of the Korean war". While the TV show M*A*S*H lasted 11 seasons, the actual Korean War barely lasted 3 years.
When Buddy (Jack Black) says, "I'm strong. And I can kick," he is imitating the actions of Crispin Glover during a notorious appearance on David Letterman.
When Buddy starts reading out Spanish phrases in the study room, he says "Dónde estás la biblioteca" and "Dónde estás al restaurante" instead of "Dónde está la biblioteca" and "Dónde está al restaurante." As opposed to the common textbook phrases "Where is the library" and "Where is the restaurant," what he says actually translates to "Where are you, library" and "Where are you, to the restaurant." Although this may seem like an accidental error, the mistakes may well have been intended to show Señor Chang's inability as a teacher to impart even the most basic elements of Spanish grammar to his students.