The Korean word kuksu corresponds to the Chinese 国手, 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.
The tournament is organized according to a usual challenger tournament, the winner of which will play the title holder in a best of three. In 2010 the title holder Lee Sedol was taking a break from professional competition and the final of the challenger tournament was decisive for the title.
YY: Kuksu literally means "Hand of the nation" (Kuk: Nation Su: Hand). It is a poetic expression refering to a strongest 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. Imagine the 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.
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.
RSM: While YY is mostly right here, I just wanted to note that Yi Changho indeed is regularly referred to as Yi Kuksu. These honorifics work more or less as a kind of nickname, derived from the title with which a player has been associated the most. For Yi Changho or Cho Hunhyeon, they are most commonly associated in Korea with the Kuksu tourney. Other very strong players are sometimes called "So-and-so kuksu" informally merely as a mark of respect, even if they had not won the Kuksu tourney. Seo Bongsu is the lone exception to this, always being referred to as "Seo Myeongin" instead, due to his exploits in that tourney, first taking the title away from Cho Namcheol as a rookie pro in 1971, and playing some of his most sensational games against Cho Hunhyeon over that title.