The Brainiacs are the members of McKinley's academic decathlon club. The club has only been mentioned in A Night of Neglect so far. Although not much has been said about the club, they are seen to be extremely successful, winning at Regionals against Carmel High School's team (which includes Sunshine Corazon) and ultimately winning Nationals in Detroit, Michigan.
At the end of the episode, the last category is revealed, which was Hermaphrodite Nazi Sympathizers. If you recall back earlier in the episode, Holly taught the kids a lesson covering this topic, which is why they knew the answer.