Kwan (hanja: 館; hangul: 관) in Korean literally means building or hall, but in the context of martial arts can also refer to a school or clan of martial artists who follow the same style or leader.
The phrase nine kwans refers to the original five schools plus four major schools that opened after the conclusion of the Korean War.
Shortly after the Korean War, at the urging of the South Korean government, the Korea Taekwondo Association (KTA) was established to consolidate and unify the kwans.
By 1978 the KTA had coordinated the Unification Proclamation, in which all nine remaining kwans agree to abide by Kukkiwon-style taekwondo and rank promotions.
Note however that many of the schools split during this period, with some factions still practicing (even today) their original martial arts styles.