The Korean word kuksu corresponds to the Chinese guoshou, a very high term of praise with a meaning like 'national treasure'. It is applied for high mastery in a very few fields such as medicine and go. There were a number of recognised kuksu in Korean when sunjang baduk was the only version of go played.
In the modern era, Kuksu is a title competed for annually, corresponding therefore in prestige to the Honinbo title in Japan as having the closest tie with the old tradition.
Winners:
YY: Kuksu literally means "Hand of the nation" (Kuk: Nation Su: Hand). It is a poetic expression refering to a stongest player in the nation and has been the highest title of honor for a Baduk player in Korea. Kuksu as a title of honor belonged to Cho Nam-ch'eol being the undisputed best player in his time. Since Cho Nam-ch'eol became inactive, the title of honor has been acknowledged for Cho Hun-hyeon. They are commonly refered as Cho Kuksu. As Cho Hun-hyeon is still active and produces good games, Yi Ch'ang-ho, though clearly the best player since the mid 90's, is not generally refered as Kuksu (or Yi Kuksu).
The winners of Kuksu title (as a tournament title) are one of the two strongest players at the time -- mostly the strongest -- with Rui Naiwei being an exception. Imagine a shock-wave Rui Naiwei, a woman and foreigner, created by winning the title and furthermore beating the two best players, Yi Ch'ang-ho and Cho Hun-hyeon, en route! Rui was honored for a Woman of the Year by the Korean government for the coup.
Grauniad: gobase.org transcribes Kuksu as "Guksu" and refers to it as a "minor" (Korean) tournament.
YY: Guksu is closer to the actual Korean pronounciation of the referent. KuksuJeon? (Jeon: competition) is only 4th in terms of prize money; however, it is most prestigeous among the Korean tournaments. It is almost inconceivable -- to Koreans, at least :o) -- that in the modern Baduk era, Kuksu as a title of honor (not a tournament title) is to be acknowledged for a player who has not been dominant in the tournament (for this reason, Yu Ch'ang-hyeok can never be refered as Kuksu (Yu Kuksu) although Kuksu as a title of honor can be acknowledged for more than one player from an era).
In March 2nd, 2004, there was an upset comparable to Rui's winning the title. Ch'oe Ch'eol-han, who did not win a single game over Yi prior to the Kuksu challenging series, beat Yi in the deciding 5th match. Yi did not resign even long after it became clear that the match was lost, which is rare (reflecting the importance of the title for him). While Yi struggled and tried desperate attempts toward the end, the commentators kept silence. It was a moment filled with pathos. Yi who always appears unaffected by match outcomes perhaps got emotional. Interestingly, Kuksu is one tournament (with a challeging series between the reigning champ and the challenger), of which Yi has not produced a long string of consecuive wins although he often expresses that winnning or defending the title is very high in priority.